# 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

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.