Every compact space is paracompact. Let X be the one-point compactification of a countably infinite discrete space X 0, and let Y be the one-point real analysis - On a paracompact space, there is a continuous. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if. (Metric spaces and compact Hausdorff spaces are paracompact. (cf. [il] and ), and every paracompact space is normal .) In §2 we prove Theorem 1, after.
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Smirnov metrization theorem A topological space is metrizable paracompact space and only if it is paracompact, Hausdorff, and locally metrizable. Michael selection theorem states that lower semicontinuous multifunctions from X into nonempty closed convex subsets of Banach paracompact space admit continuous selection iff X is paracompact.
Although a product of paracompact spaces need not be paracompact, the following are true: The product of a paracompact space and a compact space is paracompact.
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The product of a metacompact space and a compact space is metacompact. Both these results can be proved by the tube lemma which is used in paracompact space proof that a product of paracompact space many compact spaces is compact.
Paracompact Hausdorff spaces[ paracompact space ] Paracompact spaces are sometimes required to paracompact space be Hausdorff to extend their properties. Every paracompact Hausdorff space is a shrinking spacethat is, every open cover of a paracompact Hausdorff space has a shrinking: This means the following: In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover see below.
This property is sometimes used to define paracompact spaces at least in the Hausdorff case. Partitions of unity are useful paracompact space 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 paracompact space well knownand this definition is then extended to the whole space via a partition of unity.
The if direction is straightforward.
It is proved that a countably paracompact paracompact space space a perfectly normal space or a monotonically normal space is strongly paracompact if and only if every increasing open cover of the space has a paracompact space open refinement.
Moreover, it is shown that a space is linearly provided that every increasing open cover of the space has a point-countable open refinement.
Introduction The strongly paracompact property has been an interesting covering property in general topology. It is a natural generalization of compact spaces.
Characterizations of Strongly Paracompact Spaces
It retains enough structure to enjoy many of the properties of compact spaces, yet sufficiently general to include a much wider class of spaces. On one hand, the strongly paracompact property is special since it is different in many aspects with other covering properties.
For example, it is not implied even by metrizability; it is not preserved under finite-to-one closed mappings; it has no -heredity. Unlike paracompact space, the strongly paracompact property has not paracompact space characterizations.
The definition of the property is based on the existence of star-finite open refinement of every open cover. It is difficult to discover strongly paracompact spaces paracompact space only such a definition.