Cover (topology)

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In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, if

C = \lbrace U_\alpha: \alpha \in A\rbrace

is an indexed family of sets U_\alpha, then C is a cover of X if

X \subseteq \bigcup_{\alpha \in A}U_{\alpha}.

Cover in topology[edit]

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 Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if

Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}

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 open cover if each of its members is an open set (i.e. each Uα 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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any xX, there exists some neighborhood N(x) of x such that the set

\left\{ \alpha \in A : U_{\alpha} \cap N(x) \neq \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. (locally finite implies point finite)

Refinement[edit]

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 \in B}

is a refinement of

U_{\alpha \in A} \qquad \mbox{when} \qquad \forall \beta \ \exists \alpha \ V_\beta \subseteq U_\alpha.

In other words, there is a refinement map \phi: B \rightarrow 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 is a preorder on the set of covers of X.

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 < b_1), 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.

Compactness[edit]

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[edit]

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.

See also[edit]

Notes[edit]

  1. ^ Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111. 
  2. ^ Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2. 

References[edit]

  1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. ISBN 0-486-40680-6
  2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

External links[edit]