σ-compact space

A topological space is called σ-compact or countable in infinity if it can be written as a countable union of compact subspaces. So σ-compactness is a weakening of the topological concept of compactness. The letter σ in the name stems from the fact that the union of sets was previously also called a sum , the name was formed analogously to " σ-finite ". A locally compact Hausdorff space is countable in infinity if and only if the infinitely distant point added in the Alexandroff compactification has a countable surrounding base .

The term is important for abstract integration theory, together with local compactness and the separation axiom T _{3, it} guarantees the existence of a compact exhaustion .

For example , a σ-compact topological space is equipped with the standard topology, because it applies so that it can be represented as a countable union of the compact topological spaces . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} =\cup _{n=1}^{\infty }[-n,n]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle [-n,n]}$

literature

• Jürgen Elstrodt : Measure and integration theory. Springer, Berlin et al. 1996, ISBN 3-540-15307-1 .
• Jürgen Heine: Topology and Functional Analysis. Basics of abstract analysis with applications. Oldenbourg, Munich et al. 2002, ISBN 3-486-24914-2 .

Individual evidence

1. ^ Boto von Querenburg: Set theoretical topology. Springer-Verlag, 2013, ISBN 978-3-642-56860-2 , p. 111 ( limited preview in Google book search).
2. ↑ Heine: Topology and Functional Analysis. 2002, p. 336.