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., \(\mathbb R^n\),
\({\rm S}^n\)).

* (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 \(A:= T \times [0,1)\), 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
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\(^\omega\) 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 1 nov 2018