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By Lee J.M.

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In other words, the every Vi is contained in some Uβ and the new cover has only a finite number of member sets that are “near” any given point. 26 A topological space X is called paracompact if it is Hausdorff and if every open cover of X has a refinement to a locally finite cover. 27 A base (or basis) for the topology of a topological space X is a collection of open B sets such that all open sets from the topology T are unions of open sets from the familyB. A topological space is called second countable if its topology has a countable base.

An M-chart on M is a homeomorphism x whose domain is some subset U ⊂ M and such that x(U ) is an open subset (in the relative topology) of a fixed model space M ⊂ B. 7 Let Γ be a C r -pseudogroup of transformations on a model space M. A Γ-atlas, for a topological space M is a family of charts AΓ = {(xα , Uα )}α∈A (where A is just an indexing set) which cover M in the sense that M = α∈A Uα and such that whenever Uα ∩ Uβ is not empty then the map xβ ◦ x−1 α : xα (Uα ∩ Uβ ) → xβ (Uα ∩ Uβ ) is a member of Γ.

The map α : G × M → M will be continuous if for any point (g0 , x0 ) ∈ G × M and any open set U ⊂ M containing α(g0 , x0 ) we can find an open set S × V containing (g0 , x0 ) such that α(S × V ) ⊂ U. Since the topology of G is discrete, it is necessary and sufficient that there is an open V such that α(g0 × V ) ⊂ U . ) are continuous for every g ∈ G. 22 Let G and M be as above. ) is continuous and such that the following hold: 1) α(g2 , α(g1 , x)) = α(g2 g1 , x) for all g1 , g2 ∈ G and all x ∈ M .

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