door space

A topological space $X$ is called a door space if every subset of $X$ is either open or closed.

From the definition, it is immediately clear that any discrete space is door.

To find more examples, let us look at the singletons of a door space $X$. For each $x\in X$, either $\{x\}$ is open or closed. Call a point $x$ in $X$ open or closed according to whether $\{x\}$ is open or closed. Let $A$ be the collection of open points in $X$. If $A=X$, then $X$ is discrete. So suppose now that $A\ne X$. We look at the special case when $X-A=\{x\}$. It is now easy to see that the topology $\tau$ generated by all the open singletons makes $X$ a door space:

Proof.

If $B\subseteq X$ does not contain $x$, it is the union of elements in $A$, and therefore open. If $x\in B$, then its complement $B^c$ does not, so is open, and therefore $B$ is closed. ∎

Since $\tau =P(A)\cup \{X\}$, the space $X$ not discrete. In addition, $X$ and $\mathrm{\varnothing }$ are the only clopen sets in $X$.

When $X-A$ has more than one element, the situation is a little more complicated. We know that if $X$ is door, then its topology $𝒯$ is strictly finer then the topology $\tau$ generated by all the open singletons. McCartan has shown that $𝒯=\tau \cup 𝒰$ for some ultrafilter in $X$. In fact, McCartan showed $𝒯$, as well as the previous two examples, are the only types of possible topologies on a set making it a door space.

Title	door space
Canonical name	DoorSpace
Date of creation	2013-03-22 18:46:11
Last modified on	2013-03-22 18:46:11
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54E99