door space

In mathematics, in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both).Kelley, ch.2, Exercise C, p. 76. The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".

Properties

Here are some facts about door spaces:

* A Hausdorff door space has at most one accumulation point.
* In a Hausdorff door space if $x$ is not an accumulation point then $\$ is open.

To prove the first assertion, let $X$ be a Hausdorff door space, and let $x \neq y$ be distinct points. Since $X$ is Hausdorff there are open neighborhoods $U$ and $V$ of $x$ and $y$ respectively such that $U \cap V = \varnothing.$ Suppose $y$ is an accumulation point. Then $U \setminus \ \cup \$ is closed, since if it were open, then we could say that $\ = (U \setminus \ \cup \) \cap V$ is open, contradicting that $y$ is an accumulation point. So we conclude that as $U \setminus \ \cup \$ is closed, $X \setminus \left(U \setminus \ \cup \\right)$ is open and hence \ = U \cap \left \setminus (U \setminus \ \cup \)\right/math> is open, implying that $x$ is not an accumulation point.

An example of a $T_$ topological space with more than one accumulation point is the $x_$ anchor topological space. The $x_$ anchor topological space is made up from a non-empty set $X$ equipped with the $\mathcal_$ topology, where $\mathcal_=\$. In such topological spaces, every point is an accumulation point except for $x_$.

See also

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References

Bibliography

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Properties of topological spaces

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