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

Covers are commonly used in the context of topology. If the set ${\displaystyle X}$ is a topological space, then a cover ${\displaystyle C}$ of ${\displaystyle X}$ is a collection of subsets ${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$ of ${\displaystyle X}$ whose union is the whole space ${\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha }}$. In this case ${\displaystyle C}$ is said to cover ${\displaystyle X}$, or that the sets ${\displaystyle U_{\alpha }}$ cover ${\displaystyle X}$.

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

A cover of ${\displaystyle X}$ is said to be locally finite if every point of ${\displaystyle X}$ has a neighborhood that intersects only finitely many sets in the cover. Formally, ${\displaystyle C=\{U_{\alpha }\}}$ is locally finite if, for any ${\displaystyle x\in X}$, there exists some neighborhood ${\displaystyle N(x)}$ of ${\displaystyle x}$ such that the set $${\displaystyle \left\{\alpha \in A:U_{\alpha }\cap N(x)\neq \varnothing \right\}}$$ is finite. A cover of ${\displaystyle X}$ is said to be point finite if every point of ${\displaystyle X}$ is contained in only finitely many sets in the cover. A cover is point finite if locally finite, though the converse is not necessarily true.

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

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

A refinement of a cover ${\displaystyle C}$ of a topological space ${\displaystyle X}$ is a new cover ${\displaystyle D}$ of ${\displaystyle X}$ such that every set in ${\displaystyle D}$ is contained in some set in ${\displaystyle C}$. Formally,

In other words, there is a refinement map ${\displaystyle \phi :B\to A}$ satisfying ${\displaystyle V_{\beta }\subseteq U_{\phi (\beta )}}$ for every ${\displaystyle \beta \in B.}$ This map is used, for instance, in the Čech cohomology of ${\displaystyle X}$.