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):= {aA | b < a < c} and I(b):= {aA | 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.