Door space

In topology, a door space is a topological space in which each subset is open or closed or both.

Examples

• A discrete space is a door space since each subset is both open and closed.
• The subset $\{0\}\cup \{1/n:n=1,2,\ldots \}$ of the real numbers with the usual topology is a door space. Any set containing the point 0 is closed: any set not containing the point 0 is open.

References

• J.L. Kelley (1955). General topology. van Nostrand, 76.