Coverage (mathematics)

In mathematics, coverage is a fundamental concept from set theory . Open coverings play an important role in the compactness of topological spaces .

Definitions

Overlap

A family of subsets of is called covering of if ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle A}$ ${\displaystyle B\subset A}$

${\displaystyle B\subset \bigcup _{i\in I}A_{i}}$

applies. The coverage is called finite (or countable ) if the index set is finite (or countable). ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle I}$

Partial coverage

Are and covers of , then is called partial cover of , if for each one exists with . ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle (C_{j})_{j\in J}}$ ${\displaystyle B}$ ${\displaystyle (C_{j})_{j\in J}}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle j\in J}$ ${\displaystyle i\in I}$ ${\displaystyle C_{j}=A_{i}}$

refinement

If and are two overlaps of , then is called finer than if there is an index for each such that it holds. The system of sets is then called refinement or refinement coverage of . ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle (D_{k})_{k\in K}}$ ${\displaystyle B\subset A}$ ${\displaystyle (D_{k})_{k\in K}}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle k\in K}$ ${\displaystyle i\in I}$ ${\displaystyle D_{k}\subset A_{i}}$ ${\displaystyle (D_{k})_{k\in K}}$ ${\displaystyle (A_{i})_{i\in I}}$

Quasi-shrinkage and shrinkage

A refinement, as defined above, is called a quasi-shrinkage, if even holds. Is in addition and for all , it is called a contraction. ${\displaystyle {\overline {D_{k}}}\subset A_{i}}$ ${\displaystyle K=I}$ ${\displaystyle {\overline {D_{i}}}\subset A_{i}}$ ${\displaystyle i\in I}$

Coverings in topological spaces

Open / closed coverage

A covering of a topological space is called open (or closed ) if all in are open (or closed). ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle A_{i}}$ ${\displaystyle X}$

compactness

→ Main article : Compact space

A topological space is called compact if every open cover of contains a finite partial cover . ${\displaystyle X}$ ${\displaystyle X}$

Overlap properties

• An overlap is called punctual if every point of the space lies in at most finitely many overlap sets. A topological space is called metacompact if every open cover has a point refinement.
• An overlap is called locally finite if every point of the space has a neighborhood that intersects at most a finite number of overlap sets. As is well known, a topological space is called paracompact if every open cover has a locally finite refinement.
• A cover is called -local finite , if it can be written as a countable union of families of sets , so that each point of space has a neighborhood for each , which cuts out at most finitely many sets . ${\displaystyle \sigma }$ ${\displaystyle \cup _{n\in \mathbb {N} }{\mathcal {A}}_{n}}$ ${\displaystyle {\mathcal {A}}_{n}}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {A}}_{n}}$
• A cover is called -discreet if it can be written as a countable union of families of sets , so that for each point and for each there is a neighborhood of this point which cuts out at most one of the sets . The -discrete and -local coverages play an important role in the Bing-Nagata-Smirnov theorem . ${\displaystyle \sigma }$ ${\displaystyle \cup _{n\in \mathbb {N} }{\mathcal {A}}_{n}}$ ${\displaystyle {\mathcal {A}}_{n}}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {A}}_{n}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

normality

A T ₁ -space is normal if and only if every open locally finite cover has a shrinkage.

See also

• Lebesgue coverage dimension

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9
• Karl Peter Grotemeyer : Topology , Bibliographisches Institut Mannheim (1969), ISBN 3-411-00836-9