Door space

Here are some facts about door spaces:

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

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

An example of a T₀ topological space with more than one accumulation point is the ${\displaystyle x_{0}}$ anchor topological space. The ${\displaystyle x_{0}}$ anchor topological space is made up from a non-empty set ${\displaystyle X}$ equipped with the ${\displaystyle {\mathcal {T}}_{x_{0}}}$ topology, where ${\displaystyle {\mathcal {T}}_{x_{0}}=\{O\in {\mathcal {P}}(X)\mid x_{0}\in O\}\cup \{\emptyset \}}$. In such topological spaces, every point is an accumulation point except for ${\displaystyle x_{0}}$.

• Clopen set – Subset which is both open and closed

• Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.