By Stein M.R., Dennis R.K. (eds.)

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Vn } such that V i ⊂ Ui for each i = 1, . . , n. Let iβα : N (β) → N (α) be a simplicial map given by the vertex transformation iβα (Vi ) = Ui for each i. Then wβi = i∗βα (vαi ) ∈ [uαi ] for each i = 1, . . , k. 36 CHAPTER I. BACKGROUND IN TOPOLOGY Let ε = mini dist(V i , X \ Ui ). We may assume without loss of generality that Ui = X for each i. Since V i ∩ X \ Ui = ∅ and V i , X \ Ui are compact, non-empty sets, we deduce that ε is a positive real number. Let Y be a compact space and let f, g: Y →X be two maps such that d(f(y), g(y)) < ε for each y ∈ Y .

Now we consider the Urysohn function α: X → [0, 1] which takes the value 0 on X \ U and 1 on K. If we set η(x, t) = (x, α(x)t) for (x, t) ∈ Z then we obtain the desired map. Now, consider the map f : Z → Y deﬁned by the conditions: f(x, 0) = f(x) for x ∈ X and f (x, t) = h(x, t) for x ∈ K and t ∈ [0, 1]. Since Y ∈ ANR and Z is closed in Z , there exists a continuous extension f of f to a neighbourhood V of Z in Z which has values in Y . Finally we deﬁne H: Z = X × [0, 1] → Y by putting H(x, t) = f (η(x, t)) for (x, 1) ∈ X × [0, 1] and we obtain the required homotopy.

Then S = i=1 Mi with the closed sets Mi and δ(Mi ) < 2. Let E n be an n-dimensional subspace of E. Then n S ∩ En = Mi ∩ E n i=1 and in view of the Lusternik–Schnirelman–Borsuk theorem (see [De3-M, p. 22] or [DG-M, p. 43]) there exists i such that the set Mi ∩ E n contains a pair of antipodal points, x and −x. Hence δ(Mi ) ≥ 2 for this i, a contradiction. e. we have α(B(x0 , r)) = 2r and β(B(x0 , r)) = r provided dim E = +∞. 13) Example. Let r: E → cl B(0, 1) be the retraction map deﬁned as follows: x if x ≤ 1, r(x) = x if x > 1.