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).
[1] The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
are two topologically indistinguishable points, the singleton
[2] So is every quotient of a door space.
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 door space is either discrete or has exactly one accumulation point.
is a space with distinct accumulations points
having respective disjoint neighbourhoods
)[4] An example of door space with more than one accumulation point is given by the particular point topology on a set
The open sets are the subsets containing a particular point
(This is a door space since every set containing
Another example would be the topological sum of a space with the particular point topology and a discrete space.
with no isolated point are exactly those with a topology of the form
[5] Such spaces are necessarily infinite.
There are exactly three types of connected door spaces