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 $\lbrace x\rbrace$ is open or closed. Call a point $x$ in $X$ open or closed according to whether $\lbrace x\rbrace$ 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 $X-A=\lbrace x\rbrace$ . 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.

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 $\mathcal{T}$ is strictly finer then the topology $\tau$ generated by all the open singletons. McCartan has shown that $\mathcal{T}=\tau \cup \mathcal{U}$ for some ultrafilter in $X$ . In fact, McCartan showed $\mathcal{T}$ , as well as the previous two examples, are the only types of possible topologies on a set making it a door space.

Bibliography

1

J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

2

S.D. McCartan, Door Spaces are identifiable, Proc. Roy. Irish Acad. Sect. A, 87 (1) 1987, pp. 13-16.