# Thm: A subspace A of R^n is compact if and only if it is cl.

Thm: A subspace A of R^n is compact if and only if it is clâ€¦ Show more It should be a variation of this proof: Thm: A subspace A of R^n is compact if and only if it is closed and is bounded in the euclidean metric d or the square metric p. Proof: It will suffice to consider only the metric p; the inequalities: p(x,y) <= d(x,y) <= sqrt(n)p(x,y) imply that A is bounded under d if and only if it is bounded under p. Suppose that A is compact. Then it is closed. Consider the collection of open sets {B_{p}(0,m) | m is an element of the positive integers} whose union is all of R^n. Some finite subcollection covers A. It follows that A is contained in B_{p}(0, M) for some M. Therefore, for any two points x and y of A, we have p(x,y) <= 2M. Thus A is bounded under p. Conversely, suppose that A is closed and bounded underp; suppose that p(x,y) < = N for every pair x, y of points in A. Choose a point x_{0} of A, and let p(x_{0}, 0) = b. The triangle inequality implies that p(x, 0) < = N+b for every x in A. If P = N + b, then A is a subset of the cue [-P,P]^n, which is compact. Being closed, A is also compact. end proof. I also know this: the proof should be a modification of that in the text, according to the following. Rather than use the equivalence of the metrics and , use the equivalence of the topologies generated by open balls and open cubes of â€œradiusâ€ . You can use these in the places Munkres uses . Furthermore, where Munkres estimates , you should instead estimate . â€¢ Show less

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