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By Gehman H. M.

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Let {ei }i∈I be a basis for the space where eij = 1 if j = i, and zero otherwise. Each ei lies in B. the smallest ball containing n basis vectors ei1 , . . , ein would n 1 eit , with radius 1 − n1 . Then each ball of diameter 2 1 − n1 can cover be centered at n t=1 at most n basis vectors. Thus it would take an infinite number of these balls to cover B. Since any set of diameter 2 1 − 1 n of B of sets with diameter 2 1 − is contained in a ball of diameter 2 1 − n1 , any covering 1 n would have to be infinite.

However in the 50’s and 60’s, it was shown that we merely require the image to be somehow closer to compact than X itself. At some points we pause from our discussion of what types of sets have the topological fixed point property to consider some results on classes of functions for which closed, bounded, convex sets that the fixed point property. The generalizations we present here will be given in the following template as to facilitate comparison. 17 Let X be a subset of a space E, and f : X → E.

We will show that x is our desired point. To this end, let Sn be a k-simplex in our complex, containing xn , and let xn0 , . . , xnk be the vertices of Sn . As n → ∞, the diameter of Sn goes to zero. Thus xni → x for each i. Let fn (xni ) = yin . So we can write xn = without loss of generality that xn → k n n i=0 λi yi , where x, λni → λi , and could use convergent subsequences. Since for each n, k i=0 λi yin = 1, λi ≥ 0. So x = → yi , yin ∈ F (xni ), k i=0 λi yi k n n i=0 λi = 1, λi ≥ 0. We can assume yin → yi all converge.

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