By Haynes R. Miller, Douglas C. Ravenel

Throughout the wintry weather and spring of 1985 a Workshop in Algebraic Topology used to be held on the college of Washington. The direction notes through Emmanuel Dror Farjoun and via Frederick R. Cohen contained during this quantity are conscientiously written graduate point expositions of convinced points of equivariant homotopy concept and classical homotopy idea, respectively. M.E. Mahowald has incorporated many of the fabric from his additional papers, signify a variety of modern homotopy concept: the Kervaire invariant, solid splitting theorems, desktop calculation of volatile homotopy teams, and reviews of L(n), Im J, and the symmetric teams.

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We prove a series of lemmas. 1 Let G be an ordinal and let 20 CHAPTER 2 fGg : g [ Gg be an open cover of a metric space X. Write /[ fGb : 0 # b , gg; Ug ¼ Gg for g [ G. Then fUg : g [ Gg is a disjoint discretely s-decomposable cover of X by F s -sets. Proof. Let d be the metric on X. For each g in G and each n $ 1, write ÿn GðnÞ g ¼ fx : Bðx; 2 Þ , Gg g; where we use Bðx; rÞ to denote Bðx; rÞ ¼ fy : dðy; xÞ , rg: Then GðnÞ g is closed for n $ 1, and Gg ¼ 1 [ GðnÞ g : n¼1 Write UgðnÞ ¼ GðnÞ g /[ fGb : 0 # b , gg and Ug ¼ 1 [ UgðnÞ : n¼1 Then each set UgðnÞ is closed and fUg : g [ Gg is a disjoint cover of X by F s -sets.

Consider the neighborhood U of xÃ deﬁned by ' & 1 Ã Ã U ¼ x : rðx; x Þ , dðx Þ : 2 Suppose that for some a; b in G with a – b, the set U meets Ga at some point a and that U meets Gb at b. Then 1 da ðaÞ , ea ðaÞ # 3 1 db ðbÞ , eb ðbÞ # 3 1 d ðaÞ; 3 b 1 d ðbÞ: 3 a Adding these inequalities, we obtain da ðaÞ þ db ðbÞ , 1 db ðaÞ þ da ðbÞ : 3 Using the Lipschitz conditions, we have da ðxÃ Þ þ db ðxÃ Þ ÿ rða; xÃ Þ ÿ rðb; xÃ Þ , Thus 1À Á 1 da ðxÃ Þ þ db ðxÃ Þ þ rða; xÃ Þ þ rðb; xÃ Þ : 3 3 A DISJOINT DISCRETELY s-DECOMPOSABLE FAMILY OF F s -SETS 35 4 2 dðxÃ Þ # da ðxÃ Þ þ db ðxÃ Þ 3 3 Á 4À , rða; xÃ Þ þ rðb; xÃ Þ 3 4 , dðxÃ Þ; 3 since a and b belong to U.

14 is like that used in proving the Tietze extension theorem. 15 is due to Hansell [19] who uses a much less direct method. Chapter 3 Selectors for upper semi-continuous functions with nonempty compact values In this chapter we obtain selectors of the ﬁrst Borel class and of the ﬁrst Baire class for certain upper semi-continuous set-valued functions with nonempty compact values. Before we state the main results it will be convenient to give some deﬁnitions. We suppose that Z is a space with a Hausdorff topology t and also a metric d, not necessarily related to the topology t.