54A05-AlexandroffSpaceIsT1IfAndOnlyIfItIsDiscrete.tex

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\pmtitle{Alexandroff space is T1 if and only if it is discrete}
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\textbf{Proposition.} Let $X$ be an Alexandroff space. Then $X$ is $\mathrm{T}_{1}$ if and only if $X$ is discrete.

\textit{Proof.} ,,$\Leftarrow$'' It is easy to see, that every discrete space is Alexandroff and $\mathrm{T}_{1}$.

,,$\Rightarrow$'' Recall that topological space is $\mathrm{T}_1$ if and only if every subset is equal to the intersection of all its open neighbourhoods. So let $x\in X$. Then the intersection of all open neighbourhoods $\{x\}^{o}$ of $x$ is equal to $\{x\}$. But since $X$ is Alexandroff, then $\{x\}^{o}=\{x\}$ is open and thus points are open. Therefore $X$ is discrete. $\square$
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