54A05-ClosureAxioms.tex

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\pmtitle{closure axioms}
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\pmtype{Definition}
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\pmsynonym{Kuratowski's closure axioms}{ClosureAxioms}
\pmsynonym{Kuratowski closure axioms}{ClosureAxioms}
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\pmdefines{closure operator}

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A \emph{closure operator} on a set $X$ is an operator which assigns a set $A^c$ to each subset $A$ of $X$, and such that the following (Kuratowski's closure axioms) hold for any subsets $A$ and $B$ of $X$:
\begin{enumerate}
\item $\emptyset^c = \emptyset$;
\item $A\subset A^c$;
\item $(A^c)^c = A^c$;
\item $(A\cup B)^c = A^c\cup B^c.$
\end{enumerate}

The following theorem due to Kuratowski says that a closure operator characterizes a unique topology on $X$:

\textbf{Theorem.} Let $c$ be a closure operator on $X$, and let $\mathcal{T} = \{X-A: A\subseteq X,\; A^c=A\}$. Then $\mathcal{T}$ is a topology on $X$, and $A^c$ is the $\mathcal{T}$-closure of $A$ for each subset $A$ of $X$.
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