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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, particularly
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, a cover of a set $X$ is a collection of sets whose union includes $X$ as a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. Formally, if $C = \lbrace U_\alpha : \alpha \in A \rbrace$ is an
indexed family In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of sets $U_\alpha,$ then $C$ is a cover of $X$ if :$X \subseteq \bigcup_U_.$

# Cover in topology

Covers are commonly used in the context of
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

. If the set $X$ is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, then a ''cover'' $C$ of $X$ is a collection of subsets $\_$ 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 $C=\_$ of $X$ whose union contains $Y$, i.e., $C$ is a cover of $Y$ if :$Y \subseteq \bigcup_U_.$ This difference between the definition of a cover of a (universal) topological space and a cover of a topological subspace must be noted. Applications in analysis effectively use the subspace definition. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(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 A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
that intersects only
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
ly 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\$ 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, :$D = \_$ is a refinement of $C = \_$ if for all $\beta \in B$ there exists $\alpha \in A$ such that $V_ \subseteq U_.$ In other words, there is a refinement map $\phi : B \to A$ satisfying $V_ \subseteq U_$ for every $\beta \in B.$ This map is used, for instance, in the Čech cohomology of $X$. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 $a_0 < a_1 < \cdots < a_n$ being $a_0 < b_0 < a_1 < a_2 < \cdots < a_ < b_1 < a_n$), considering
topologies s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...

(the
standard topology of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification ...
in euclidean space being a refinement of the
trivial topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
). When subdividing
simplicial complex In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
es (the first
barycentric subdivision In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedron, tetrahedra, or, in general, a convex polytope into simplex, simplices with the same dimension, b ...

of a simplicial complex is a refinement), the situation is slightly different: every
simplex In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

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 In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''. The general definition makes sense for arbitrary coverings and does ...
.

# 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 $\mathcal$ be a topological basis of $X$ and $\mathcal$ be an open cover of $X.$ First take $\mathcal = \.$ Then $\mathcal$ is a refinement of $\mathcal$. Next, for each $A \in \mathcal,$ we select a $U_ \in \mathcal$ containing $A$ (requiring the axiom of choice). Then $\mathcal = \$ is a subcover of $\mathcal.$ 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 Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
: 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
subcover, (or equivalently that every open cover has a countable refinement); ;
MetacompactIn mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open Refinement (topology), refinement. That is, given any open cover of the topological space, there is a refineme ...
: if every open cover has a point-finite open refinement; ;
Paracompact In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: 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 In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topological invariant, topologically invariant way. Informal discussion Fo ...
''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. If no such minimal ''n'' exists, the space is said to be of infinite covering dimension.

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Atlas (topology) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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Covering space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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Partition of a set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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Set cover problemThe set cover problem is a classical question in combinatorics Combinatorics is an area of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ...
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Star refinement In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''. The general definition makes sense for arbitrary coverings and does ...
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Grothendieck topologyIn category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is call ...

# References

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