Paracompact Topological Spaces

In General
* History: Invented by J Dieudonné, it is important because it introduces metric methods in general topology.
$ Def: A (Hausdorff) topological space such that 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)].

Examples
* (0) Any compact space, of course.
* (1) A Hausdorff, locally compact manifold expressible as a countable union of compact subsets (e.g., Rⁿ, Sⁿ).
* (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, aka the Alexandrov line.
$ Def: If T:= {countable ordinal numbers}, then A:= T × [0,1), totally ordered by the lexicographic order (t₁, x₁) < (t₂, x₂) if t₁< t₂ or t₁ = t₂ and x₁ < x₂.
* 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^ structure.
@ References: Kneser AASF(58); in Hocking & Young 61.

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 11 aug 2007