54A05-InteriorAxioms.tex

\documentclass[12pt]{article}
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\pmtitle{interior axioms}
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\pmtype{Definition}
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\pmdefines{interior operator}

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\newtheorem{axiom}{Axiom}
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\begin{document}
Let $S$ be a set.  Then an \emph{interior operator} is a function
$\,^\circ \colon \mathcal{P}(S) \to \mathcal{P}(S)$ which satisfies the 
following properties:
\begin{axiom}
$S^\circ = S$
\end{axiom}
\begin{axiom}
For all $X \subset S$, one has $X^\circ \subseteq S$.
\end{axiom}
\begin{axiom}
For all $X \subset S$, one has $(X^\circ)^\circ = X^\circ$.
\end{axiom}
\begin{axiom}
For all $X, Y \subset S$, one has $(X \cap Y)^\circ = 
X^\circ \cap Y^\circ$.
\end{axiom}

If $S$ is a topological space, then the operator which assigns to
each set its interior satisfies these axioms.  Conversely, given an
interior operator $\,^\circ$ on a set $S$, the set $\{X^\circ \mid
X \subset S\}$ defines a topology on $S$ in which $X^\circ$ is the
interior of $X$ for any subset $X$ of $S$.  Thus, specifying an
interior operator on a set is equivalent to specifying a topology
on that set.

The concepts of interior operator and closure operator are closely
related.  
Given an interior operator $\,^\circ$, one can 
define a closure operator $\,^c$ by the condition
 \[ X^c = ({(X')^\circ})\vphantom{X}' \]
and, given a closure operator $\,^c$, one can 
define an interior operator $\,^\circ$ by the condition
\ \[ X^\circ = ({(X')^c})\vphantom{X}' .\]
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\end{document}