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 $\mathcal{T}$ 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.
References
- 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.
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 |