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Homedoor space

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# 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 $A\neq X$. 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 $\varnothing$ 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.

## Mathematics Subject Classification

54E99*no label found*

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