54A05-AlexandroffSpace.tex

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\pmtitle{Alexandroff space}
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\pmtype{Definition}
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\begin{document}
Topological space $X$ is called \textit{Alexandroff} if the intersection of every family of open sets is open.

Of course every finite topological space is Alexandroff, but there are also bigger Alexandroff spaces. For example let $\mathbb{R}$ denote the set of real numbers and let $\tau=\{[a,\infty)\ |\ a\in\mathbb{R}\} \cup \{(b,\infty)\ |\ b\in\mathbb{R}\}$. Then $\tau$ is a topology on $\mathbb{R}$ and $(\mathbb{R},\tau)$ is an Alexandroff space.

If $X$ is an Alexandroff space and $A\subseteq X$, then we may talk about smallest open neighbourhood of $A$. Indeed, let $$A^{o}=\bigcap \{U\subseteq X\ |\ U\mbox{ is open and }A\mbox{ is contained in }U\}.$$
Then $A^{o}$ is open.
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