Cover (topology) explained

X is a family of subsets of X whose union is all of X . More formally, if C = \lbrace U _{\alpha} : \alpha \in A \rbrace is an indexed family of subsets U _{\alpha\subset} X (indexed by the set A ), then C is a cover of X if c up _{\alpha} U _{\alpha} \supseteq X . Thus the collection \lbrace U _{\alpha} : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U _{\alpha} .

A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set.

Cover in topology

Covers are commonly used in the context of topology. If the set

X is a topological space, then a cover C of X is a collection of subsets \{U _{\alpha\}} _{\alpha\in} of X whose union is the whole space X . In this case we say that C covers X , or that the sets U _{\alpha} cover X .

Also, if

Y is a (topological) subspace of X , then a cover of Y is a collection of subsets C = \{U _{\alpha\}} _{\alpha\in} of X whose union contains Y , i.e., C is a cover of Y ifY \subseteq c up _{\alpha} U _{\alpha} .

That is, we may cover

Y with either sets in Y itself or sets in the parent space X .

Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X.

We say that C is an if each of its members is an open set (i.e. each U_α is contained in T, where T is the topology on X).

A cover of X is said to be locally finite if every point of X has a neighborhood that intersects only finitely many sets in the cover. Formally, C = is locally finite if for any

x \in X, there exists some neighborhood N(x) of x such that the set\left\{ \alpha \in A : U _{\alpha} \cap N(x) ≠ \varnothing \right\}

is finite. A cover of X is said to be point finite if every point of X is contained in only finitely many sets in the cover. A cover is point finite if it is locally finite, though the converse is not necessarily true.

Refinement

A refinement of a cover

C of a topological space X is a new cover D of X such that every set in D is contained in some set in C . Formally,D = \{ V _{\beta} \} _{\beta} is a refinement of C = \{ U _{\alpha} \} _{\alpha} if for all \beta \in B there exists \alpha \in A such that V _{\beta} \subseteq U _{\alpha} .

In other words, there is a refinement map

\phi : B \to A satisfying V _{\beta} \subseteq U _{\phi(\beta)} for every \beta \in B. This map is used, for instance, in the Čech cohomology of X .^[1]

Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are in the cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in the cover.

The refinement relation on the set of covers of

X is transitive, irreflexive, and asymmetric.

Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an interval (one refinement of

a _{0} < a _{1} < … < a _{n} being a _{0} < b _{0} < a _{1} < a _{2} < … < a _{n-1} < b _{1} < a _{n} ), considering topologies (the standard topology in Euclidean space being a refinement of the trivial topology). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra.

Yet another notion of refinement is that of star refinement.

Subcover

A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let

l{B} be a topological basis of X and l{O} be an open cover of X. First take l{A} = \{ A \in l{B} : thereexists U \in l{O} suchthat A \subseteq U \}. Then l{A} is a refinement of l{O} . Next, for each A \in l{A}, we select a U _{A} \in l{O} containing A (requiring the axiom of choice). Then l{C} = \{ U _{A} \in l{O} : A \in l{A} \} is a subcover of l{O}. Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is Lindelöf.

Compactness

The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be

Compact

if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);

Lindelöf

if every open cover has a countable subcover, (or equivalently that every open cover has a countable refinement);

Metacompact

if every open cover has a point-finite open refinement;

Paracompact

if every open cover admits a locally finite open refinement.

For some more variations see the above articles.

Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for which this is true.^[2] If no such minimal n exists, the space is said to be of infinite covering dimension.

References

1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999.
2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

Notes and References

1. Book: Bott, Tu. Differential Forms in Algebraic Topology. 1982. 111.
2. Book: Munkres , James . James Munkres . 1999 . Topology . 2nd . . 0-13-181629-2.

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