# $F_{\sigma}$ set

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 $G_{\delta}$ set (http://planetmath.org/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:

 $X=\{(x,0)\mid x\in\mathbb{R}\}\cup\bigcup_{r\in\mathbb{Q}}\{(ry,y)\mid y\in% \mathbb{R}\}$
Title $F_{\sigma}$ set FsigmaSet 2013-03-22 14:37:59 2013-03-22 14:37:59 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 54A05 G_DeltaSet G_deltaSet PavedSet PavedSpace