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54A20-ContinuityAndConvergentNets.tex
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54A20-ContinuityAndConvergentNets.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ContinuityAndConvergentNets}
\pmcreated{2013-03-22 18:37:53}
\pmmodified{2013-03-22 18:37:53}
\pmowner{azdbacks4234}{14155}
\pmmodifier{azdbacks4234}{14155}
\pmtitle{continuity and convergent nets}
\pmrecord{5}{41369}
\pmprivacy{1}
\pmauthor{azdbacks4234}{14155}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{54A20}
%\pmkeywords{net}
%\pmkeywords{directed set}
%\pmkeywords{continuous function}
%\pmkeywords{convergence}
\pmrelated{Net}
\pmrelated{Continuous}
\endmetadata
%packages
\usepackage{amsmath,mathrsfs,amsfonts,amsthm}
%theorem environments
\theoremstyle{plain}
\newtheorem*{thm*}{Theorem}
\newtheorem*{lem*}{Lemma}
\newtheorem*{cor*}{Corollary}
\newtheorem*{prop*}{Proposition}
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\begin{document}
\begin{thm*}
Let $X$ and $Y$ be topological spaces. A function $f:X\rightarrow Y$ is continuous at a point $x\in X$ if and only if for each net $(x_\alpha)_{\alpha\in A}$ in $X$ converging to $x$, the net $(f(x_\alpha))_{\alpha\in A}$ converges to $f(x)$.
\end{thm*}
\begin{proof}
If $f$ is continuous, $(x_\alpha)_{\alpha\in A}$ converges to $x$, and $V$ is an open neighborhood of $f(x)$ in $Y$, then $f^{-1}(V)$ is an open neighborhood of $x$ in $X$, so there exists $\alpha_0\in A$ such that $x_\alpha\in f^{-1}(V)$ for $\alpha\geq\alpha_0$. It follows that $f(x_\alpha)\in V$ for $\alpha\geq\alpha_0$, hence that $f(x_\alpha)\rightarrow f(x)$. Conversely, suppose there exists a net $(x_\alpha)_{\alpha\in A}$ in $X$ converging to $x$ such that $(f(x_\alpha))_{\alpha\in A}$ does not converge to $f(x)$, so that, for some open subset $V$ of $Y$ containing $f(x)$ and every $\alpha_0\in A$, there exists $\alpha\geq\alpha_0\in A$ such that $f(x_{\alpha})\notin V$, hence such that $x_\alpha\notin f^{-1}(V)$; as $x_\alpha\rightarrow x$ by hypothesis, this implies that $f^{-1}(V)$ cannot be a neighborhood of $x$, and thus that $f$ fails to be continuous at $x$.
\end{proof}
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\end{document}