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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 general consensus abo ...
, particularly
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
, a cover of a set X is a collection of sets whose union includes X as a
subset 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). ...

subset
. 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 s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
. 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 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). I ...
(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 of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval ...
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, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. Preorders are more general than equivalence relations and (non-strict) partial ...
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 ...

topologies
(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 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 \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 space, 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 space, Compact: if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement); ;Lindelöf space, Lindelöf: if every open cover has a countable subcover, (or equivalently that every open cover has a countable refinement); ;metacompact space, Metacompact: if every open cover has a point-finite open refinement; ;paracompact space, 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. If no such minimal ''n'' exists, the space is said to be of infinite covering dimension.


See also

* Atlas (topology) * Covering space * Partition of a set * Set cover problem * Star refinement * Grothendieck topology


Notes


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.


External links

* {{springer, title=Covering (of a set), id=p/c026950 Topology General topology