Cover (topology)

In mathematics, and more particularly in set theory, a cover (or covering)Template:R of a set $X$ is a family of subsets of $X$ whose union is all of $X$. More formally, if $C=\{U_{\alpha }:\alpha \in A\}$ is an indexed family of subsets $U_{\alpha }\subset X$ (indexed by the set $A$), then $C$ is a cover of $X$ if $$\bigcup _{\alpha \in A}{U}_{\alpha }=X.$$ Thus the collection $\{U_{\alpha }:\alpha \in A\}$ is a cover of $X$ if each element of $X$ belongs to at least one of the subsets $U_{\alpha }$.

Definition

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 $\{U_{\alpha }{\}}_{\alpha \in A}$ of $X$ whose union is the whole space $X=\bigcup _{\alpha \in A}{U}_{\alpha }$. In this case $C$ is said to cover $X$, or that the sets $U_{\alpha }$ cover $X$.Template:R

If $Y$ is a (topological) subspace of $X$, then a cover of $Y$ is a collection of subsets $C=\{U_{\alpha }{\}}_{\alpha \in A}$ of $X$ whose union contains $Y$. That is, $C$ is a cover of $Y$ if $$Y\subseteq \bigcup _{\alpha \in A}{U}_{\alpha }.$$ Here, $Y$ may be covered with either sets in $Y$ itself or sets in the parent space $X$.

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, $C=\{U_{\alpha }\}$ is locally finite if, for any $x\in X$, there exists some neighborhood $N(x)$ of $x$ such that the set $$\{\alpha \in A:U_{\alpha }\cap N(x)\ne \varnothing \}$$ 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.Template:R A cover is point finite if locally finite, though the converse is not necessarily true.

Subcover

Let $C$ be a cover of a topological space $X$. A subcover of $C$ is a subset of $C$ that still covers $X$. The cover $C$ is said to be an Template:Vanchor if each of its members is an open set. That is, each $U_{\alpha }$ is contained in $T$, where $T$ is the topology on $X$.Template:R

A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let $ℬ$ be a topological basis of $X$ and $𝒪$ be an open cover of $X$. First, take $𝒜=\{A\in ℬ:\text{ there exists }U\in 𝒪\text{ such that }A\subseteq U\}$. Then $𝒜$ is a refinement of $𝒪$. Next, for each $A\in 𝒜,$ one may select a $U_A\in 𝒪$ containing $A$ (requiring the axiom of choice). Then $𝒞=\{U_A\in 𝒪:A\in 𝒜\}$ is a subcover of $𝒪.$ Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence, second countability implies space is Lindelöf.

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,

$D=\{V_{\beta }{\}}_{\beta \in B}$ is a refinement of $C=\{U_{\alpha }{\}}_{\alpha \in A}$ 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 $\varphi :B\to A$ satisfying $V_{\beta }\subseteq U_{\varphi (\beta )}$ for every $\beta \in B.$ This map is used, for instance, in the Čech cohomology of $X$.Template:R

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 and reflexive, i.e. a Preorder. It is never asymmetric for $X\ne \mathrm{\varnothing }$.

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 $a_0<a_1<\cdots <a_n$ being $a_0<b_0<a_1<a_2<\cdots <a_{n-1}<b_1<a_n$), 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.Template:Citation needed

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; and
• orthocompact: if every open cover has an interior-preserving 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.Template:R If no such minimal $n$ exists, the space is said to be of infinite covering dimension.

See also

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References

• Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. Template:ISBN
• Template:Kelley 1975

External links

• Template:Springer