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In [[mathematics]], a '''topological space''' is an ordered pair <math>(X,O)</math> where <math>X</math> is a set and <math>O</math> is a certain collection of subsets of <math>X</math> called the <i>open sets</i> or the <i>topology</i> of <math>X</math>. The topology of <math>X</math> introduces a structure on the set <math>X</math> which is useful for defining some important abstract notions such as the "closeness" of two elements of <math>X</math> and [[convergence of sequences]] of elements of <math>X</math>.
In [[mathematics]], a '''topological space''' is an [[ordered pair]] <math>(X,\mathcal T)</math> where <math>X</math> is a set and <math>\mathcal T</math> is a certain collection of subsets of <math>X</math> called the <i>open sets</i> or the <i>topology</i> of <math>X</math>. The topology of <math>X</math> introduces an abstract structure of space in the set <math>X</math>, which allows to define general notions such as of a point being surrounded by a set (by a neighborhood) or belonging to its boundary, of [[convergence of sequences]] of elements of <math>X</math>, of [[connectedness]], of a space or set being [[contractible space|contractible]], etc.


==Definition==  
==Definition==  
A topological space of open sets is an ordered pair <math>(X,O)</math> where <math>X</math> is a set and <math>O</math> is a collection of subsets of <math>X</math> (i.e., any element <math> A \in O </math> is a subset of ''X'') with the following three properties:
A topological space is an ordered pair <math>(X,\mathcal T)</math> where <math>X</math> is a set and <math>\mathcal T</math> is a collection of subsets of <math>X</math> (i.e., any element <math> A \in \mathcal T </math> is a subset of ''X'') with the following three properties:


# <math>X</math> and <math>\emptyset</math> (the empty set) are in <math>O</math>
# <math>X</math> and <math>\varnothing</math> (the empty set) are in <math>\mathcal T</math>
# The union of any family (infinite or otherwise) of elements of <math>O</math> is again in <math>O</math>     
# The union of any family (infinite or otherwise) of elements of <math>\mathcal T</math> is again in <math>\mathcal T</math>     
# The intersection of finitely many elements of <math>O</math> is again in <math>O</math>   
# The intersection of two elements of <math>\mathcal T</math> is again in <math>\mathcal T</math>   


Elements of the set <math>O</math> are called open sets of <math>X</math>.  We often simply writte <math>X</math> instead of <math>(X,O)</math> once the topology <math>O</math> is established.
Elements of the set <math>\mathcal T</math> are called open sets of <math>X</math>.  We often simply write <math>X</math> instead of <math>(X,\mathcal T)</math> once the topology <math>\mathcal T</math> is established.


Once we have a topology of open sets on <math>X</math>, we define the ''closed sets'' of <math>X</math> to be the compliments (in <math>X</math>) of the open sets; the closed sets of <math>X</math> have the properties that
Once we have a topology in <math>X</math>, we define the ''closed sets'' of <math>X</math> to be the [[complement (set theory)|complement]]s (in <math>X</math>) of the open sets; the closed sets of <math>X</math> have the following characteristic properties:


# <math>X</math> and <math>\emptyset</math> (the empty set) are closed
# <math>X</math> and <math>\varnothing</math> (the empty set) are closed
# The intersection of any family of closed sets is closed
# The intersection of any family of closed sets is closed
# The union of finitely many closed sets is closed
# The union of two closed sets is closed


Alternatively, notice that we could have defined a topology of closed sets (having the properties above as axioms) and defined the open sets relative to that topology as compliments of closed sets. The set of opens relative to a topology of closed sets then obeys the axioms for a topology of opens, and therefore closed and these two kinds of topologies occur simultaneously. Similarly, the axioms for a topology of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology of opens, and conversely.  Thus, the notion of topology is a relative one, and these equivalences allow one to study the same topological structure from a number of different viewpoints simultaneously.
Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as complements of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set <math>X</math> we obtain a one to one correspondence between topologies in <math>X</math> and systems of neighborhoods in <math>X</math>.  These correspondences allow one to study the topological structure from different viewpoints.


==The category of topological spaces==
==The category of topological spaces==


Given that topological capture notions of geometry, a good notion of isomorphism in the category of topological spaces should require that equivalent spaces have equivalent topologies.  The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.  
Given that topological spaces capture notions of geometry, a good notion of isomorphism in the [[category]] of topological spaces should require that equivalent spaces have equivalent topologies.  The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.  


A function <math>f:X\to Y</math> is ''continuous'' if <math>f^{-1}(V)</math> is open in <math>X</math> for every open in <math>Y</math>. Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if <math>f^{-1}(Z)</math> is closed in <math>X</math> for each closed subset <math>Z</math> of Y, then <math>f</math> is continuous in the sense just defined, and conversely.  
A function <math>f:X\to Y</math> is ''continuous'' if <math>f^{-1}(V)</math> is open in <math>X</math> for every open in <math>Y</math>. Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if <math>f^{-1}(Z)</math> is closed in <math>X</math> for each closed subset <math>Z</math> of Y, then <math>f</math> is continuous in the sense just defined, and conversely.  


Isomorphisms in the category of topological spaces (often referred to as a ''homeomorphism'') are bijective and continuous with continuous inverses.
Isomorphisms in the category of topological spaces (often referred to as a [[homeomorphism]]) are bijective and continuous with continuous inverses.
 
The category of topological spaces has a number of nice properties; there is an initial object (the empty set),  subobjects (the subspace topology) and quotient objects (the quotient topology), and products and coproducts exist as well. The necessary topologies to define on the latter two objects become clear immediately; if they're going to be universal in the category of topological spaces, then the topologies should be the coarsest making the canonical maps commute.


== Examples ==
== Examples ==
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:<math> \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.</math>
:<math> \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.</math>


Then a topology <math>O</math> can be defined on <math>X=\mathbb{R}</math> to consist of <math>\emptyset</math> and all sets of the form:  
Then a topology <math>\mathcal T</math> can be defined on <math>X=\mathbb{R}</math> to consist of <math>\emptyset</math> and all sets of the form:  


:<math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math>
:<math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math>
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where <math>\Gamma</math> is any arbitrary index set, and <math>a_{\gamma}</math> and <math>b_{\gamma}</math> are real numbers satisfying <math>a_\gamma < b_\gamma</math> for all <math>\gamma \in \Gamma </math>. This is the familiar topology on <math>\mathbb{R}</math> and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set <math>X</math> and in the next example another topology on <math>\mathbb{R}</math>, albeit a relatively obscure one,  will be constructed.  
where <math>\Gamma</math> is any arbitrary index set, and <math>a_{\gamma}</math> and <math>b_{\gamma}</math> are real numbers satisfying <math>a_\gamma < b_\gamma</math> for all <math>\gamma \in \Gamma </math>. This is the familiar topology on <math>\mathbb{R}</math> and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set <math>X</math> and in the next example another topology on <math>\mathbb{R}</math>, albeit a relatively obscure one,  will be constructed.  


2. Let <math>X=\mathbb{R}</math> as before. Let <math>O</math> be a collection of subsets of <math>\mathbb{R}</math> defined by the requirement that <math>A \in O </math> if and only if <math>A=\emptyset</math> or <math>A</math> contains all except at most a finite number of real numbers. Then it is straightforward to verify that <math>O</math> defined in this way has the three properties required to be a topology on <math>\mathbb{R}</math>. This topology is known as the ''cofinite topology'' or ''Zariski topology''.
2. Let <math>X=\mathbb{R}</math> as before. Let <math>\mathcal T</math> be a collection of subsets of <math>\mathbb{R}</math> defined by the requirement that <math>A \in \mathcal T </math> if and only if <math>A=\emptyset</math> or <math>A</math> contains all except at most a finite number of real numbers. Then it is straightforward to verify that <math>\mathcal T</math> defined in this way has the three properties required to be a topology on <math>\mathbb{R}</math>. This topology is known as the ''cofinite topology'' or ''Zariski topology''.


3. Every [[metric space|metric]] <math>d</math> on <math>X</math> gives rise to a topology on <math>X</math>. The open ball with centre <math>x \in X</math> and radius <math>r > 0</math> is defined to be the set
3. Every [[metric space|metric]] <math>d</math> on <math>X</math> gives rise to a topology on <math>X</math>. The open ball with centre <math>x \in X</math> and radius <math>r > 0</math> is defined to be the set
:<math> B_r(x) = \{ y \in X \mid d(x,y) < r \}. </math>
:<math> B_r(x) = \{ y \in X \mid d(x,y) < r \}. </math>
A set <math>A \subset X</math> is open if and only if for every <math>x \in A</math>, there is an open ball with centre <math>x</math> contained in <math>A</math>. The resulting topology is called the topology induced by the metric <math>d</math>. The standard topology on <math>\mathbb{R}</math>, discussed in Example 1, is induced by the metric <math>d(x,y) = |x-y|</math>.
A set <math>A \subset X</math> is open if and only if for every <math>x \in A</math>, there is an open ball with centre <math>x</math> contained in <math>A</math>. The resulting topology is called the topology induced by the metric <math>d</math>. The standard topology on <math>\mathbb{R}</math>, discussed in Example 1, is induced by the metric <math>d(x,y) = |x-y|</math>.
4. For a given set <math>X</math>, the family <math>\mathcal{T} = \{ \emptyset, X \}</math> is a topology: the ''[[indiscrete space|indiscrete]]'' or ''weakest'' topology.
5. For a given set <math>X</math>, the family <math>\mathcal{T} = \mathcal{P}X</math> of all subsets of <math>X</math> is a topology: the ''[[discrete space|discrete]]'' topology.


== Neighborhoods ==
== Neighborhoods ==
Given a topological space <math>(X,O)</math> of opens, we say that a subset <math>N</math> of <math>X</math> is a ''neighborhood'' of a point <math>x \in X</math> if <math>N</math> contains an open set <math>U \in O</math> containing the point <math>x</math> <ref>Some authors use a different definition, in which a neighborhood ''N'' of ''x'' is an open set containing ''x''.</ref>  
Given a topological space <math>(X,\mathcal T)</math> of opens, we say that a subset <math>N</math> of <math>X</math> is a ''neighborhood'' of a point <math>x \in X</math> if <math>N</math> contains an open set <math>U \in \mathcal T</math> containing the point <math>x</math> <ref>Some authors use a different definition, in which a neighborhood ''N'' of ''x'' is an open set containing ''x''.</ref>  


If <math>N_x</math> denotes the system of neighborhoods of <math>x</math> relative to the topology <math>O</math>, then the following properties hold:
If <math>N_x</math> denotes the system of neighborhoods of <math>x</math> relative to the topology <math>\mathcal T</math>, then the following properties hold:
# <math>N_x</math> is not empty for any <math>x \in X</math>
# <math>N_x</math> is not empty for any <math>x \in X</math>
# If <math>U</math> is in <math>N_x</math> then <math>x \in U</math>
# If <math>U</math> is in <math>N_x</math> then <math>x \in U</math>
# The intersection of two elements of <math>N_x</math> is again in <math>N_x</math>
# The intersection of two elements of <math>N_x</math> is again in <math>N_x</math>
# If <math>U</math> is in <math>N_x</math> and <math>V \subset X</math> contains <math>U</math>, then <math>V</math> is again in <math>N_x</math>
# If <math>U</math> is in <math>N_x</math> and <math>V \subset X</math> contains <math>U</math>, then <math>V</math> is again in <math>N_x</math>
# If <math>U</math> is in <math>N_x</math> then there exists a <math>V \in N_x</math> such that <math>V</math> is a subset of <math>U</math> and <math>V \in N_y</math> for all <math>y \in V</math>
# If <math>U</math> is in <math>N_x</math> then there exists a <math>V \in N_x</math> such that <math>V</math> is a subset of <math>U</math> and <math>U \in N_y</math> for all <math>y \in V</math>


Conversely, if we define a topology of neighborhoods on <math>X</math> via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from: <math>U</math> is open if it is in <math>N_x</math> for all <math>x \in U</math>. Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from.  All this just means that a given topological spaces is the same, regardless of which axioms we choose to start from.  
Conversely, if we define a topology of neighborhoods on <math>X</math> via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from: <math>U</math> is open if it is in <math>N_x</math> for all <math>x \in U</math>. Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from.  All this just means that a given topological space is the same, regardless of which axioms we choose to start from.  


The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the <math>I</math>-adic topology on a ring <math>A</math> is Hausdorff if and only if <math>\bigcap I^n=0</math>, thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.
The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the <math>I</math>-adic topology on a ring <math>A</math> is Hausdorff if and only if <math>\bigcap I^n=0</math>, thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.
==Bases and sub-bases==
A '''basis''' for the topology <math>\mathcal T</math> on ''X'' is a collection <math>\mathcal B</math> of open sets such that every open set is a union of elements of <math>\mathcal B</math>.  For example, in a metric space the open balls form a basis for the metric topology.  A '''sub-basis''' <math>\mathcal S</math> is a collection of open sets such that the finite intersections of elements of <math>\mathcal S</math> form a basis for <math>\mathcal T</math>.


== Some topological notions==
== Some topological notions==
This section introduces some important topological notions. Throughout, ''X'' will denote a topological space with the topology ''O''.  
This section introduces some important topological notions. Throughout, <math>X</math> will denote a topological space with the topology <math>\mathcal T</math>.  


; Partial list of topological notions  
; Partial list of topological notions  
; Limit point : A point <math>x \in X</math> is a limit point of a subset ''A'' of ''X'' if any open set in ''O'' containing ''x'' also contains a point <math>y \in A</math> with <math>y \ne x</math>. An equivalent definition is that <math>x \in X</math> is a limit point of ''A'' if every neighbourhood of ''x'' contains a point <math>y \in A</math> different from ''x''.
; Open cover : A collection <math>\mathcal{U}</math> of open sets of ''X'' is said to be an open cover for ''X'' if each point <math>x \in X</math> belongs to at least one of the open sets in <math>\mathcal{U}</math>
; Path: A path <math>\gamma</math> is a [[continuous function]] <math>\gamma:[0,1]\rightarrow X</math>. The point <math>\gamma(0)</math> is said to be the '''starting point''' of <math>\gamma</math> and <math>\gamma(1)</math> is said to be the '''end point'''. A path joins its starting point to its end point
; Hausdorff/separability property: ''X'' has the Hausdorff (or separability) property if for any pair <math>x,y \in X</math> there exist ''disjoint'' sets ''U'' and ''V'' with <math>x \in U</math> and <math>y \in V</math>
; Connectedness: ''X'' is connected if given any two ''disjoint'' open sets ''U'' and ''V'' such that <math>X=U \cup V </math>, then either ''X=U'' or ''X=V''
; Path-connectedness : ''X'' is path-connected if for any pair <math>x,y \in X</math> there exists a path joining ''x'' to ''y''
; Compactness : ''X'' is said to be compact if any open cover of ''X'' has a ''finite sub-cover''. That is, any open cover has a finite number of elements which again constitute an open cover for ''X'' 


A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.
;Closure : {{Main|Closure (topology)}}The ''[[Closure (topology)|closure]]'' in <math>X</math> of a subset <math>E</math> is the [[intersection]] of all closed sets of <math>X</math> which contain <math>E</math> as a subset.
;Interior : {{Main|Interior (topology)}}The ''[[Interior (topology)|interior]]'' in <math>X</math> of a subset <math>E</math> is the [[union]] of all open sets of <math>X</math> which are contained in <math>E</math> as a subset.
; Limit point : {{Main|Limit point}}A point <math>x \in X</math> is a ''[[limit point]]'' of a subset <math>A</math> of <math>X</math> if any open set in <math>\mathcal T</math> containing <math>x</math> also contains a point <math>y \in A</math> with <math>y \ne x</math>. An equivalent definition is that <math>x \in X</math> is a limit point of <math>A</math> if every neighbourhood of <math>x</math> contains a point <math>y \in A</math> different from <math>x</math>.
; Open cover : A collection <math>\mathcal{U}</math> of open sets of <math>X</math> is said to be an ''[[open cover]]'' for <math>X</math> if each point <math>x \in X</math> belongs to at least one of the open sets in <math>\mathcal{U}</math>.
; Path: A ''path'' <math>\gamma</math> is a [[continuous function]] <math>\gamma:[0,1]\rightarrow X</math>. The point <math>\gamma(0)</math> is said to be the '''starting point''' of <math>\gamma</math> and <math>\gamma(1)</math> is said to be the '''end point'''. A path joins its starting point to its end point.
 
; Hausdorff/separability property: {{Main|Separation axioms}}<math>X</math> has the ''[[Hausdorff space|Hausdorff]]'' (or separability, or T2) property if for any pair <math>x,y \in X</math> there exist ''disjoint'' sets <math>U</math> and <math>V</math> with <math>x \in U</math> and <math>y \in V</math>.
 
; Noetherianity: {{Main|Noetherian space}}<math>X</math> is ''[[Noetherian space|noetherian]]'' if it satisfies the [[descending chain condition]] for closed set: any descending chain of closed subsets <math>Y_0\supseteq Y_1\supseteq\ldots</math> is eventually stationary; i.e., if there is an index <math>i\geq 0</math> such that <math>Y_{i+r}=Y_i</math> for all <math>r\geq 0</math>.
 
; Connectedness: {{Main|Connected space}}<math>X</math> is ''[[Connected space|connected]]'' if given any two ''disjoint'' open sets <math>U</math> and <math>V</math> such that <math>X=U \cup V </math>, then either <math>X=U</math> or <math>X=V</math>.
; Path-connectedness: <math>X</math> is ''[[Path-connected space|path-connected]]'' if for any pair <math>x,y \in X</math> there exists a path joining <math>x</math> to <math>y</math>. A path connected topological space is also connected, but the converse need not be true.
 
; Compactness: {{Main|Compactness axioms}}<math>X</math> is said to be ''[[Compact space|compact]]'' if any open cover of <math>X</math> has a ''finite sub-cover''. That is, any open cover has a finite number of elements which again constitute an open cover for <math>X</math>.
 
A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space.


==Induced topologies==
==Induced topologies==
A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as '''induced topologies'''. Descriptions of some induced topologies are given below. Throughout, <math>(X,O_X)</math> will denote a topological space.
A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as '''induced topologies'''. Descriptions of some induced topologies are given below. Throughout, <math>(X,\mathcal T_X)</math> will denote a topological space.


; Some induced topologies  
===Some induced topologies===
; Relative topology : If ''A'' is a subset of ''X'' then open sets may be defined on ''A'' as sets of the form <math>O \cap A</math> where ''O'' is any open set in <math>O_X</math>. The collection of all such open sets defines a topology on ''A'' called the ''relative topology'' of ''A'' as a subset of ''X''    
====Relative topology====
; Quotient topology : If ''Y'' is another set and ''q'' is a surjective function from ''X'' to ''Y'' then open sets may be defined on ''Y'' as subsets ''U'' of ''Y'' such that <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in O_X</math>. The collection of all such open sets defines a topology on ''Y'' called the ''quotient topology'' induced by ''q''
{{Main|Subspace topology}}
If <math>A</math> is a subset of <math>X</math> then open sets may be defined on <math>A</math> as sets of the form <math>U \cap A</math> where <math>U</math> is any open set in <math>\mathcal T_X</math>. The collection of all such open sets defines a topology on <math>A</math> called the ''relative topology'' of <math>A</math> as a subset of <math>X</math>    
====Quotient topology====
{{main|Quotient topology}}
If <math>Y</math> is another set and <math>q</math> is a surjective function from <math>X</math> to <math>Y</math> then open sets may be defined on <math>Y</math> as subsets <math>U</math> of <math>Y</math> such that <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in \mathcal T_X</math>. The collection of all such open sets defines a topology on <math>Y</math> called the ''quotient topology'' induced by <math>q</math>.
====Product topology====
{{Main|Product topology}}
If <math>(X_\lambda,\mathcal{T}_\lambda)_{\lambda\in\Lambda}</math> is a family of topological spaces, then the ''product topology'' on the [[Cartesian product]] <math>\prod_{\lambda\in\Lambda} X_\lambda</math> has as sub-basis the sets of the form <math>\prod_{\lambda\in\Lambda} U_\lambda</math> where each <math>U_\lambda \in \mathcal{T}_\lambda</math> and <math>U_\lambda = X_\lambda</math> for all but finitely many <math>\lambda\in\Lambda</math>.


== See also ==
== See also ==
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# K. Yosida, ''Functional Analysis'' (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
# K. Yosida, ''Functional Analysis'' (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
# D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [http://www.maths.tcd.ie/~dwilkins/Courses/212/]  
# D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [http://www.maths.tcd.ie/~dwilkins/Courses/212/]


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In mathematics, a topological space is an ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is a certain collection of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} called the open sets or the topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} introduces an abstract structure of space in the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , which allows to define general notions such as of a point being surrounded by a set (by a neighborhood) or belonging to its boundary, of convergence of sequences of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , of connectedness, of a space or set being contractible, etc.

Definition

A topological space is an ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is a collection of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} (i.e., any element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal T } is a subset of X) with the following three properties:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varnothing} (the empty set) are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T}
  2. The union of any family (infinite or otherwise) of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T}
  3. The intersection of two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T}

Elements of the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} are called open sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . We often simply write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T)} once the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is established.

Once we have a topology in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , we define the closed sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to be the complements (in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ) of the open sets; the closed sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} have the following characteristic properties:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varnothing} (the empty set) are closed
  2. The intersection of any family of closed sets is closed
  3. The union of two closed sets is closed

Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as complements of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} we obtain a one to one correspondence between topologies in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and systems of neighborhoods in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . These correspondences allow one to study the topological structure from different viewpoints.

The category of topological spaces

Given that topological spaces capture notions of geometry, a good notion of isomorphism in the category of topological spaces should require that equivalent spaces have equivalent topologies. The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.

A function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to Y} is continuous if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(V)} is open in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} for every open in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} . Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(Z)} is closed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} for each closed subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} of Y, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous in the sense just defined, and conversely.

Isomorphisms in the category of topological spaces (often referred to as a homeomorphism) are bijective and continuous with continuous inverses.

The category of topological spaces has a number of nice properties; there is an initial object (the empty set), subobjects (the subspace topology) and quotient objects (the quotient topology), and products and coproducts exist as well. The necessary topologies to define on the latter two objects become clear immediately; if they're going to be universal in the category of topological spaces, then the topologies should be the coarsest making the canonical maps commute.

Examples

1. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{R}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.}

Then a topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} can be defined on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{R}} to consist of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} and all sets of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is any arbitrary index set, and and are real numbers satisfying for all . This is the familiar topology on and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set and in the next example another topology on , albeit a relatively obscure one, will be constructed.

2. Let as before. Let be a collection of subsets of defined by the requirement that if and only if or contains all except at most a finite number of real numbers. Then it is straightforward to verify that defined in this way has the three properties required to be a topology on . This topology is known as the cofinite topology or Zariski topology.

3. Every metric on gives rise to a topology on . The open ball with centre and radius is defined to be the set

A set is open if and only if for every , there is an open ball with centre contained in . The resulting topology is called the topology induced by the metric . The standard topology on , discussed in Example 1, is induced by the metric .

4. For a given set , the family is a topology: the indiscrete or weakest topology.

5. For a given set , the family of all subsets of is a topology: the discrete topology.

Neighborhoods

Given a topological space of opens, we say that a subset of is a neighborhood of a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} contains an open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in \mathcal T} containing the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} [1]

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} denotes the system of neighborhoods of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} relative to the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} , then the following properties hold:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} is not empty for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X}
  2. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U}
  3. The intersection of two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x}
  4. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \subset X} contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x}
  5. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} then there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \in N_x} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in N_y} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in V}

Conversely, if we define a topology of neighborhoods on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is open if it is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U} . Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from. All this just means that a given topological space is the same, regardless of which axioms we choose to start from.

The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} -adic topology on a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is Hausdorff if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap I^n=0} , thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.

Bases and sub-bases

A basis for the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} on X is a collection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B} of open sets such that every open set is a union of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B} . For example, in a metric space the open balls form a basis for the metric topology. A sub-basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal S} is a collection of open sets such that the finite intersections of elements of form a basis for .

Some topological notions

This section introduces some important topological notions. Throughout, will denote a topological space with the topology .

Partial list of topological notions
Closure
For more information, see: Closure (topology).

The closure in of a subset is the intersection of all closed sets of which contain as a subset.

Interior
For more information, see: Interior (topology).

The interior in of a subset is the union of all open sets of which are contained in as a subset.

Limit point
For more information, see: Limit point.

A point is a limit point of a subset of if any open set in containing also contains a point with . An equivalent definition is that is a limit point of if every neighbourhood of contains a point different from .

Open cover
A collection of open sets of is said to be an open cover for if each point belongs to at least one of the open sets in .
Path
A path is a continuous function . The point is said to be the starting point of and is said to be the end point. A path joins its starting point to its end point.
Hausdorff/separability property
For more information, see: Separation axioms.

has the Hausdorff (or separability, or T2) property if for any pair there exist disjoint sets and with and .

Noetherianity
For more information, see: Noetherian space.

is noetherian if it satisfies the descending chain condition for closed set: any descending chain of closed subsets is eventually stationary; i.e., if there is an index such that for all .

Connectedness
For more information, see: Connected space.

is connected if given any two disjoint open sets and such that , then either or .

Path-connectedness
is path-connected if for any pair there exists a path joining to . A path connected topological space is also connected, but the converse need not be true.
Compactness
For more information, see: Compactness axioms.

is said to be compact if any open cover of has a finite sub-cover. That is, any open cover has a finite number of elements which again constitute an open cover for .

A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space.

Induced topologies

A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as induced topologies. Descriptions of some induced topologies are given below. Throughout, will denote a topological space.

Some induced topologies

Relative topology

For more information, see: Subspace topology.

If is a subset of then open sets may be defined on as sets of the form where is any open set in . The collection of all such open sets defines a topology on called the relative topology of as a subset of

Quotient topology

For more information, see: Quotient topology.

If is another set and is a surjective function from to then open sets may be defined on as subsets of such that . The collection of all such open sets defines a topology on called the quotient topology induced by .

Product topology

For more information, see: Product topology.

If is a family of topological spaces, then the product topology on the Cartesian product has as sub-basis the sets of the form where each and for all but finitely many .

See also

Notes

  1. ↑ Some authors use a different definition, in which a neighborhood N of x is an open set containing x.

References

  1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
  2. D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [1]
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