interior axiomsInteriorAxioms2006-12-25 19:25:432006-12-25 23:15:07Definitioninteriorinterior operator% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}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
\ \[ X^\circ = ({(X')^c})\vphantom{X}' .\]