A second course in general topology by Heikki Junnila

By Heikki Junnila

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Example Let d be a pseudometric of X and let r > s > 0. For every A ⊂ X, the family {Bd (x, s) : x ∈ A} is a partial refinement of the family {Bd (x, r) : x ∈ A}. Moreover, the cover {Bd (x, s) : x ∈ X} of X is a refinement of the cover {Bd (x, r) : x ∈ X}. 5 Definition A cover N of X is a point-star refinement of a cover L of X provided that the family {St(x, N ) : x ∈ X} is a refinement of L. The space X is fully normal provided that every open cover of X has an open point- star refinement. Example Let d be a pseudometric of X and let r > 0.

Then there exists a partition of unity {gα : α ∈ A} of X such that we have Supp(gα ) ⊂ Supp(fα ) for every α ∈ A and the family {Supp(gα ) : α ∈ A} is locally finite (as an indexed family). Proof. By Lemma 6, the function h = supα∈A fα is continuous. It follows that, for every α ∈ A, the set Uα = {x ∈ X : fα (x) > 1 h(x)} 2 is open. Note that the family U = {Uα : α ∈ A} covers X. We show that U is locally finite. Let x ∈ X. Then there exists αx ∈ A such that fαx (x) > 34 h(x). Denote by V the nbhd {z ∈ X : fαx (z) > 34 h(x)} of x.

Let ǫ > 0. Then there exists a finite cover K ⊂ N of X such that we have d(f (K)) ≤ ǫ for every K ∈ K. By Corollary 8, we have that K ∩ F = ∅ for every ˇ : K ∈ K} of N ˇ covers UN . Now, let F ∈ UN , and this means that the finite subfamily {K ˇ Then we have that K ∈ F and K ∈ H and hence that fˇ(F ) ∈ f (K) K ∈ K and F , H ∈ K. and fˇ(H) ∈ f (K). Since d(f (K)) ≤ ǫ, it follows that we have |fˇ(F ) − fˇ(H)| ≤ ǫ. We have shown that fˇ is Nˇ -uniformly continuous. 17 Lemma Let X be a Tihonov space.

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