Symbols:Greek/Sigma/Countability
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Countability
- $\sigma$
Used to denote the property of countability.
The $\LaTeX$ code for \(\sigma\) is \sigma
.
Sigma-Algebra
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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Sigma-Compact Space
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 $T$.
$F_\sigma$ Set
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 $T$.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma: 2.