PlanetMath: $F_\sigma$ set

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$F_\sigma$ set

(Definition)

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.

For instance, the set of all points in the plane such that either or is rational is an $F_\sigma$ set because it can be expressed as the union of a countable set of lines:

$\displaystyle X = \{(x,0) \mid x \in \mathbb{R} \} \cup \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\}$

" $F_\sigma$ set" is owned by rspuzio.

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See Also: $G_\delta$ set, $G_\delta$ set

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Cross-references: lines, rational, plane, points, complement, closed sets, collection, countable, union, topological space, subset
There are 2 references to this entry.

This is version 6 of $F_\sigma$ set, born on 2004-09-24, modified 2004-09-25.
Object id is 6215, canonical name is F_sigmaSet.
Accessed 2292 times total.

Classification:

AMS MSC:

54A05 (General topology :: Generalities :: Topological spaces and generalizations )

Pending Errata and Addenda

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