By Heikki Junnila

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Greatly considered as a vintage of contemporary arithmetic, this accelerated model of Felix Klein's celebrated 1894 lectures makes use of modern thoughts to ascertain 3 recognized difficulties of antiquity: doubling the amount of a dice, trisecting an attitude, and squaring a circle. contemporary scholars will locate this quantity of specific curiosity in its solutions to such questions as: less than what conditions is a geometrical building attainable?

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Note that to prove Theorem 4, we did not need any assumptions or results on the existence of continuous real-valued functions (such as the Tihonov property or the Urysohn Lemma). Therefore it is interesting to note that Theorem 4 gives as a corollary the 44 following strong result on the existence of continuous real-valued functions. 5 Tietze-Urysohn Extension Theorem Let X be a normal space, let S ⊂c X and let f : S → I be continuous. Then f has a continuous extension f¯ : X → I. Proof. The formula d(x, y) = |f (x) − f (y)| defines a totally bounded and continuous pseudometric d of S.

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.

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. In the next chapter, we use continuous pseudometrics to study paracompact spaces. 1. Construction of pseudometrics. To start with, we recall some terminology, notation and results dealing with metrics and pseudometrics.