refinement (topology)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more particularly in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a cover (or covering) of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
X is a collection of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, wher ...
of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_.


Cover in topology

Covers are commonly used in the context of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. If the set X is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, 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_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the parent space X. 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, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
(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, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
that intersects only
finite Finite is the opposite of 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 or marke ...
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 and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...
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 (the
standard topology In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
in euclidean space being a refinement of the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
). When subdividing
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es (the first
barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...
of a simplicial complex is a refinement), the situation is slightly different: every
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
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 or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
: 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, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
subcover, (or equivalently that every open cover has a countable refinement); ; Metacompact: if every open cover has a point-finite open refinement; ;
Paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
: 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 topologically invariant way. Informal discussion For ordinary Euclide ...
''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

* * * * * * *


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