In mathematics, specifically in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".

Properties and examples

Every door space is T0 are two topologically indistinguishable is neither open nor closed).

Every subspace of a door space is a door space. So is every quotient of a door space.

Every topology finer than is also a door topology.

Every discrete space is a door space. These are the spaces without accumulation point, that is, whose every point is an isolated point.

with exactly one accumulation point (and all the other point isolated) is a door space (since subsets consisting only of isolated points are open, and subsets containing the accumulation point are closed). Some examples are: (1) the one-point compactification of a discrete space (also called Fort space), where the point at infinity is the accumulation point; (2) a space with the excluded point topology, where the "excluded point" is the accumulation point.

Every Hausdorff having respective disjoint neighbourhoods)

An example of door space with more than one accumulation point is given by the particular point topology is closed.) Another example would be the topological sum of a space with the particular point topology and a discrete space.

for some free ultrafilter Such spaces are necessarily infinite.

:

• a space with the excluded point topology;

• a space with the included point topology;

• is a free ultrafilter

See also

Notes

References