In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. (Paracompact spaces are often required to be Hausdorff, but we will not make that assumption in this article.) Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

Definitions of relevant terms

• A cover of a set X is a collection of subsets of X whose union is X. In symbols, if U = {U_α : α in A} is an indexed family of subsets of X, then U is a cover iff

• A cover of a topological space X is open if all its members are open sets. In symbols, a cover U is an open cover if U is contained in T, where T is the topology on X.
• A refinement of a cover of a space X is a new cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols, the cover V = {V_β : β in B} is a refinement of the cover U = {U_α : α in A} iff, for any V_β in V, there exists some U_α in U such that V_β is contained in U_α.
• An open cover of a space X is locally finite if every point of the space has a neighborhood which intersects only finitely many sets in the cover. In symbols, U = {U_α : α in A} is locally finite iff, for any x in X, there exists some neighbourhood V(x) of x such that the set

is finite.

Note the similarity between the definitions of compact and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite". Both of these changes are significant: if we take the above definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases. This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...

Examples and counterexamples

• Every compact space is paracompact.
• Every metric space (hence, every metrisable space) is paracompact.
• Every locally compact second-countable space is paracompact.
• Every regular Lindelöf space is paracompact.
• The real line in the lower limit topology is paracompact, even though it is neither compact, locally compact, second countable, nor metrisable.

As you might guess from the generality of most of the examples above, it is actually harder to think of spaces that are not paracompact than to think of spaces that are. The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.) Another counterexample is a product of uncountably many copies of an infinite discrete space. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ... In topology, the long line is a topological space analogous to the real line, but much longer. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, an uncountable set is a set which is not countable. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

Most mathematicians who use point set topology, rather than investigate it in its own right, regard nonparacompact spaces as pathological. For example, manifolds are usually defined to be paracompact, thus allowing integration of differential forms to be defined, while excluding the long line, which is useless in almost every application. In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ...

Properties

• Every paracompact Hausdorff space is normal.
• Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact.
• A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.)
• (Smirnov metrisation theorem) A topological space is metrisable if and only if it is paracompact, Hausdorff, and locally metrisable.

In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...

Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unity relative to any open cover. This means the following: if X is a paracompact Hausdorff space with a given open cover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that: In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

• for every function f: X → R from the collection, there is an open set U from the cover such that f is identically 0 outside of U;
• for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in the collection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity. 0 (zero), alternatively called naught or nought, is both a number and a numeral. ... Look up one in Wiktionary, the free dictionary. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This page is about a higher mathematics topic. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Variations

There are several mild variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:

• Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that contain the point. In symbols, the star of x in U = {U_α : α in A} is

The notation for the star is not standardised in the literature, and this is just one possibility.

• A star refinement of a cover of a space X is a new cover of the same space such that, given any point in the space, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a star refinement of U = {U_α : α in A} iff, for any x in X, there exists a U_α in U, such that V^*(x) is contained in U_α.
• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in the cover. In symbols, U is pointwise finite iff, for any x in X, the set

is finite.

A topological space X is metacompact if every open cover has an open pointwise finite refinement, and fully normal if every open cover has an open star refinement. The adverb "countably" can be added to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to countable open covers. In mathematics the term countable set is used to describe the size of a set, e. ...

As you might guess from the terminology, a fully normal space is normal. Any space that is fully normal must be paracompact, and any space that is paracompact must be metacompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T₄ space (that is, a fully normal space that is also T₁; see Separation axioms) is the same thing as a paracompact Hausdorff space. In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ...