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
\(x_1 < x_2\).
* 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 22 sep 2019