A second course in general topology by Heikki Junnila

By Heikki Junnila

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Then we have that N = X. Let x ∈ X N . Then Nx ∈ N ˆ. N It follows from Propositions 9-11 that if a T1 -space X has a normal closed base, then X has a Hausdorff compactification and hence X is Tihonov. We shall next show that the converse holds: every Tihonov space has a normal closed base. The zero-set of a function f : X → R is the set f −1 {0}. A zero-set of a space X is the zero-set of a continuous function X → R. We denote by ZX the collection of all zero-sets of X. If f : X → R is continuous, then so is the function g, defined by the rule g(x) = min(1, |f (x)|).

Then d can be extended to a continuous pseudometric of X provided that there exists a continuous pseudometric ρ of X such that we have d ≤ ρ on A2 . Proof. Assume that ρ is as above. Define r : X 2 → R by setting r(x, y) = d(x, y), if (x, y) ∈ A2 ρ(x, y), if (x, y) ∈ A2 , and note that we have r ≤ ρ on X 2 . For all x, y ∈ X, denote by δ(x, y) the infimum of the numbers n i=0 r(xi , xi+1 ), where n ∈ N, xi ∈ X for each i, x0 = x and xn+1 = y. It is easy to see that δ is a pseudometric of X. Moreover, since r ≤ ρ, we have that δ ≤ ρ and hence δ is continuous.

Let d be a pseudometric of X, and let A ⊂ X. We denote by d(A) the d-diameter sup{d(x, y) : x, y ∈ A} of A. For x ∈ X and ∅ = A ⊂ X, we denote by d(x, A) the d-distance inf{d(x, a) : a ∈ A} of the point x to the set A. If X is a space and d is a continuous pseudometric of X, then for every non-empty A ⊂ X, the function x → d(x, A) is continuous. This is a consequence of the inequality |d(x, A) − d(y, A)| ≤ d(x, y) which is valid for all x, y ∈ X. We mention some ways of obtaining “new pseudometrics from old”.

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