By Gunnar Carlsson, Ralph Cohen, Haynes R. Miller, Douglas C. Ravenel

Those are lawsuits of a world convention on Algebraic Topology, held 28 July via 1 August, 1986, at Arcata, California. The convention served partly to mark the twenty fifth anniversary of the magazine *Topology* and sixtieth birthday of Edgar H. Brown. It preceded ICM 86 in Berkeley, and used to be conceived as a successor to the Aarhus meetings of 1978 and 1982. a few thirty papers are integrated during this quantity, often at a learn point. topics comprise cyclic homology, H-spaces, transformation teams, genuine and rational homotopy idea, acyclic manifolds, the homotopy thought of classifying areas, instantons and loop areas, and intricate bordism.

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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.