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, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.
The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact.[1] Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact.
Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.
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U=\{U\alpha:\alpha\inA\}
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X\subseteqcup\alphaU\alpha.
A cover of a topological space
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V=\{V\beta:\beta\inB\}
U=\{U\alpha:\alpha\inA\}
V\beta
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U\alpha
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V\beta\subseteqU\alpha
An open cover of a space
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U=\{U\alpha:\alpha\inA\}
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\left\{\alpha\inA:U\alpha\capV ≠ \varnothing\right\}
is finite. A topological space
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This definition extends verbatim to locales, with the exception of locally finite: an open cover
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Some examples of spaces that are not paracompact include:
Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to F-sigma subspaces as well.
Although a product of paracompact spaces need not be paracompact, the following are true:
Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compact spaces is compact.
Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.
The most important feature of paracompact Hausdorff spaces is that they admit partitions of unity subordinate 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 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 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.
There is a similarity between the definitions of compactness and paracompactness:For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the 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.
Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.
Paracompactness is similar to compactness in the following respects:
It is different in these respects:
There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:
A topological space is:
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.
Every paracompact space is metacompact, and every metacompact space is orthocompact.
U*(x):=
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| U\alpha\nix |
U\alpha.
The notation for the star is not standardised in the literature, and this is just one possibility.
\left\{\alpha\inA:x\inU\alpha\right\}
As the names imply, a fully normal space is normal and a fully T4 space is T4. Every fully T4 space is paracompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompact Hausdorff space.
Without the Hausdorff property, paracompact spaces are not necessarily fully normal. Any compact space that is not regular provides an example.
A historical note: fully normal spaces were defined before paracompact spaces, in 1940, by John W. Tukey.[8] The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces full normality and paracompactness are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later Ernest Michael gave a direct proof of the latter fact andM.E. Rudin gave another, elementary, proof.