Paracompact
Topological Spaces |

**In General**

* __History__: The concept was invented by
J Dieudonné, and is important because it introduces
metric methods in general topology.

$ __Def__: A (Hausdorff)
topological space is paracompact if every open cover has a locally finite refinement
(not necessarily a subcover).

* __Properties__: (1) They
admit a partition of unity and (2) a Riemannian metric; (3) They are always
second countable (conversely, either 2 or 3 implies
that the manifold is paracompact), (4) normal (Dieudonné),
and (5) triangulable [@ Whitehead AM(40)].

> __Online resources__:
see Wikipedia page.

**Examples**

* (0) Any compact space, of course.

* (1) A Hausdorff, locally compact
manifold expressible as a countable union of compact subsets (e.g., \(\mathbb R\)^{n},
S^{n}).

* (2) A metrizable space [@
Stone BAMS(48)].

* (3) The direct
limit of a sequence of compact spaces.

**Non-Paracompact Manifold: The Long Line**

* __Idea__: A smooth connected
non-paracompact 1D manifold, a.k.a. the Alexandrov line.

$ __Def__: If *T*:= {countable ordinal numbers}, then *A*:= *T* ×
[0,1), totally ordered by the lexicographic order (*t*_{1}, *x*_{1}) < (*t*_{2}, *x*_{2})
if *t*_{1}< *t*_{2} or *t*_{1} = *t*_{2} and *x*_{1} < *x*_{2}.

* __Basis for the topology__: *I*(*b*, *c*):=
{*a* ∈ *A* | *b* < *a *< *c*}
and *I*(*b*):= {*a* ∈ *A* | *a* > *b*}.

* __Properties__: Its definition requires the axiom of choice; It has
a non-unique C^{ω} structure.

@ __References__: Kneser AASF(58); in Hocking & Young 61.

> __Online resources__:
see Wikipedia page.

**General References**

@ __Articles__: Dieudonné JMPA(44);
Marathe JDG(72).

@ __Texts__: in Kelley 55; in Dugundji 60; in Kobayashi & Nomizu 69.

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send feedback and suggestions to bombelli at olemiss.edu – modified 16 feb 2016