Door space


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. 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.
Every subspace of a door space is a door space. So is every quotient of a door space.
Every topology finer than a door topology on a set 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.
Every space with exactly one accumulation point is a door space. Some examples are: the one-point compactification of a discrete space, where the point at infinity is the accumulation point; a space with the excluded point topology, where the "excluded point" is the accumulation point.
Every Hausdorff door space is either discrete or has exactly one accumulation point.
An example of door space with more than one accumulation point is given by the particular point topology on a set with at least three points. The open sets are the subsets containing a particular point together with the empty set. The point is an isolated point and all the other points are accumulation points. Another example would be the topological sum of a space with the particular point topology and a discrete space.
Door spaces with no isolated point are exactly those with a topology of the form for some free ultrafilter on Such spaces are necessarily infinite.
There are exactly three types of connected door spaces :