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By Heikki Junnila

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Example text

Moreover, g has the same zero-set as f . It follows that all zero-sets of X can be obtained as zero-sets of continuous functions X → I. Note that a zero-set is always closed. If Y is a metric space, then every closed subset F of Y is a zero-set: F is the zero-set of the continuous function y → d(y, F ), where d is a compatible metric for Y . Note also that if f : X → Z is continuous, then for every zero-set S of Z, the set f −1 (S) is a zero-set of X. It follows from the preceding observations that for all f, h ∈ C(X), the set {x ∈ X : f (x) ≤ h(x)} is a zero-set.

Show that the closures of F and H in βX are disjoint. [Hint: Write F = f −1 {0} and H = h−1 {0}, where f, h ∈ C(X, I). ] 31 6. A space X is extremally disconnected if G ⊂◦ X for every G ⊂◦ X. Show that X is extremally disconnected iff any two disjoint open subsets of X have disjoint closures. 7. Show that an extremally disconnected T3 -space has no non-trivial convergent sequences. [Hint: Show first that if Z is regular, z1 , z2 , ... ] 32 III CONTINUOUS PSEUDOMETRICS In this chapter, we consider the construction and existence of continuous pseudometrics on topological spaces.

We shall now exhibit some consequence’s of Michael’s Theorem. 6 Corollary Let X be a fully normal space, let F ⊂c X and let Y be a Banach space and f a continuous mapping F → Y . Then f can be extended to a continuous mapping 54 X → H, where H is the closed convex hull of f (F ). Proof. We define ϕ : X → P(Y ) be setting ϕ(x) = f {x} for x ∈ F and ϕ(x) = H for x∈X (X F . For every G ⊂◦ Y with G ∩ H = ∅, we have that {x ∈ X : ϕ(x) ∩ G = ∅} = F ) ∪ f −1 (G) and this set is open in X since f is continuous.

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