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A course in simple-homotopy theory by M.M. Cohen

By M.M. Cohen

Cohen M.M. A path in simple-homotopy idea (Springer, [1973)(ISBN 3540900551)

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Consider all collections C of subsets of X such that: (1) AEC => {xa} is frequently in A; and (2) A, BEC => A n BEe. For example, C = {X} is such a collection. Order the family of all such collections C by inclusion. , satisfies (1) and (2). By the Maximality Principle, there is a maximal such collection Co. Let Po = {(A,a)EC o x PIXaEA} and order Po by (B, /3) 2:: (A, a) <=> B c A and /3 2:: a. This gives a partial order on Po making Po into a directed set. Map Po -+ P by taking (A, a) to a. This is clearly final and thus defines a subnet we shall denote by {X(A,a)}' We claim that this subnet is universal.

Show that K2 ~P2#P2. I. General Topology 44 Figure 1-3. The torus (left) and Klein bottle (right). 6. Consider the real line R, with the equivalence relation x ~ y=x - y is rational. Show that RI ~ has an uncountable number of points, but its topology is the trivial one. 7. Consider the real line R and the integers Z. Let A = R/Z (the identification of the subspace Z to a point). ) in the upper half plane all tangent to the real line at the origin. ) in the upper half plane all tangent to the real line at the origin.

Let U be an open covering of I. Put S = {sEII [0, s] is covered by a finite subcollection of U}. Let b the least upper bound of S. Clearly S must be an interval of the form S = [0, b) or S = [0, b]. In the former case, however, consider a set U E U containing the point b. This set must contain an interval of the form [a, b]. But then we can throw U in with the hypothesized finite cover of [0, a] to obtain a finite cover of [0, b]. Thus we must have that S = [0, b] for some bE[0, 1]. But if b < 1, then a similar argument shows that there is a finite cover of [0, c] for some c > b, contradicting the choice of b.

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