Symbols:Greek/Sigma/Countability

Countability

$\sigma$

Used to denote the property of countability.

The $\LaTeX$ code for $\sigma$ is \sigma .

Let $X$ be a set.

Let $\Sigma$ be a system of subsets of $X$.

$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:

$(\text {SA 1})$	$:$	Unit:		$\ds X \in \Sigma $
$(\text {SA 2})$	$:$	Closure under Complement:	$\ds \forall A \in \Sigma:$	$\ds \relcomp X A \in \Sigma $
$(\text {SA 3})$	$:$	Closure under Countable Unions:	$\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$	$\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma $

Let $T = \struct {S, \tau}$ be a topological space.

$T$ is $\sigma$-compact if and only if $S$ is the union of the underlying sets of countably many compact subspaces of $T$.

Let $T = \struct {S, \tau}$ be a topological space.

An $F_\sigma$ set ($F$-sigma set) is a set which can be written as a countable union of closed sets of $T$.

\((\text {SA 1})\)	$:$	Unit:		\(\ds X \in \Sigma \)
\((\text {SA 2})\)	$:$	Closure under Complement:	\(\ds \forall A \in \Sigma:\)	\(\ds \relcomp X A \in \Sigma \)
\((\text {SA 3})\)	$:$	Closure under Countable Unions:	\(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\)	\(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \)