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.

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