Relative sequence compactness

The relative sequential compactness is a term used in topology , a branch of mathematics . It combines the two properties of sequence compactness and relative compactness and thus provides the existence of accumulation points in the topological closure.

definition

A topological space is given . A subset is said to be relatively sequence- compact if every sequence of elements of has a convergent subsequence with a limit value in its topological closure . ${\displaystyle (X,\tau )}$ ${\displaystyle A\subset X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle {\overline {A}}}$

Clarifications

In order to express which topology or which concept of convergence is used, a corresponding concept is sometimes used. Thus, for example, one speaks of weakly relatively sequence-compact sets when it is a question of weak convergence , or of vague relatively sequence-compact sets when it is a question of vague convergence .

sentences

• The Bolzano-Weierstrass theorem characterizes the relatively compact subsets of real and complex numbers.
• The selection set of Helly characterizes the relatively vague sequentially compact amounts of distribution functions and of finite dimensions on (with the respectively matching vague concept of convergence vague convergence of distribution functions or vague convergence (measure theory) ). ${\displaystyle \mathbb {R} }$
• The set of Prokhorov characterizes the weak relatively sequentially compact amounts of radon measurements by tight families of moderation .
• For the weak topology on Banach spaces , relative sequence compactness and relative compactness coincide according to the Eberlein – Šmulian theorem .

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .