By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

As a part of the clinical task in reference to the seventieth birthday of the Adam Mickiewicz college in Poznan, a world convention on algebraic topology used to be held. within the ensuing lawsuits quantity, the emphasis is on huge survey papers, a few offered on the convention, a few written consequently.

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8. The most useful aspect of the Arzelà–Ascoli Theorem is the following corollary. 13 Let X, d be a compact metric space and Y, d be a metric space. Every sequence ( f n )n∈N of maps from X to Y : 1. that is equicontinuous, 2. and such that, for every x ∈ X , there is a compact subset K x of Y such that, for every n ∈ N, f n (x) ∈ K x , has a uniformly convergent subsequence. Again, the second condition is equivalent to the fact that { f n (x) | n ∈ N} has a sequentially compact closure. Proof Let K be the closure of { f n | n ∈ N} in the spaces of all maps from X to Y with the sup metric sup X d .

In a metric space, one can define sequentially closed sets in a number of d , the open equivalent ways. 5 Open balls in R2 . ball with center x and radius d(x, y) < . 2 In R with distance |x − y|, the open ball of center x and radius r is the open interval (x − r, x + r ). 5, for various metrics (the gray areas, without the boundaries). 3 (Sequentially closed) Let X, d be a metric space. For every subset F of X , the following are equivalent: 1. F is sequentially closed; d centered 2. for every point x ∈ X that is not in F, there is an open ball Bx,< at x that does not meet F; 3.

We claim xn 34 A first tour of topology: metric spaces that x − = x + . For every > 0, find n 0 ∈ N such that, for all m, m n0, . As |xm − xm | < . So, for every n n 0 , xn+ − xn− = supm,m n (xm − x m ) > 0 is arbitrary, x n+ −xn− inf >0 = 0. It follows x + −x − xn+ −xn− 0, whence x − = x + . Let x = x − = x + . We must finally show that x is the limit of (xn )n∈N . n 0 , |xm − x m | < /2. For every > 0, find n 0 such that, for all m, m − + − + Then xn 0 − x n 0 /2, and xn 0 x xn 0 . It follows that x − xn−0 /2, so /2 < for every m n 0 , and xn+0 − x /2, so x m − x /2 < x − xm for every m n 0 .