$F_\sigma$ setF_sigmaSet2004-09-24 02:41:212004-09-25 01:02:35Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA 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:
$$X = \{(x,0) \mid x \in \mathbb{R} \} \cup \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\}$$