54A05-FsigmaSet.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{FsigmaSet}
\pmcreated{2013-03-22 14:37:59}
\pmmodified{2013-03-22 14:37:59}
\pmowner{rspuzio}{6075}
\pmmodifier{rspuzio}{6075}
\pmtitle{$F_\sigma$ set}
\pmrecord{9}{36215}
\pmprivacy{1}
\pmauthor{rspuzio}{6075}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{54A05}
\pmrelated{G_DeltaSet}
\pmrelated{G_deltaSet}
\pmrelated{PavedSet}
\pmrelated{PavedSpace}

\endmetadata

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\begin{document}
A subset of a topological space is called a $F_\sigma$ set if it equals the union of a countable collection of closed sets.  

The complement of a $F_\sigma$ set is a \PMlinkname{$G_\delta$ set}{G_DeltaSet}.

For instance, the $X$ set of all points $(x,y)$ in the plane such that either $y = 0$ or $x/y$ is rational is an $F_\sigma$ set because it can be expressed as the union of a countable set of lines:
 $$X = \{(x,0) \mid x \in \mathbb{R} \} \cup \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\}$$
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\end{document}