54A05-TheCategoryOfT0AlexandroffSpacesIsEquivalentToTheCategoryOfPosets.tex

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\pmtitle{the category of T0 Alexandroff spaces is equivalent to the category of posets}
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\begin{document}
Let $\mathcal{AT}$ be the category of all $\mathrm{T}_{0}$, Alexandroff spaces and continuous maps between them. Furthermore let $\mathcal{POSET}$ be the category of all posets and order preserving maps.

\textbf{Theorem.} The categories $\mathcal{AT}$ and $\mathcal{POSET}$ are equivalent.

\textit{Proof.} Consider two functors:
$$T:\mathcal{AT}\to\mathcal{POSET};$$
$$S:\mathcal{POSET}\to \mathcal{AT},$$
such that $T(X,\tau)=(X,\leq)$, where $\leq$ is an induced partial order on an Alexandroff space and $T(f)=f$ for continuous map. Analogously, let $S(X,\leq)=(X,\tau)$, where $\tau$ is an induced Alexandroff topology on a poset and $S(f)=f$ for order preserving maps. One can easily show that $T$ and $S$ are well defined. Furthermore, it is easy to verify that equalities $T\circ S=1_{\mathcal{POSET}}$ and $S\circ T=1_{\mathcal{AT}}$ hold, which completes the proof. $\square$

\textbf{Remark.} Of course every finite topological space is Alexandroff, thus we have very nice ,,interpretation'' of finite $\mathrm{T}_{0}$ spaces - finite posets (since functors $T$ and $S$ do not change set-theoretic properties of underlying sets such as finitness).
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