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By Alexandroff P.

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Note that the metric used for Rq in the Heine–Borel Theorem must be the standard one, or the result may fail even for an equivalent metric. For example, if we use the metric d(x, y) = |x − y|(1 + |x − y|)−1 on R, then all of R is closed and bounded but not compact. This furnishes another example of a closed and bounded set that is not compact. 10. If (X, d) is a compact metric space and f : X → (Z, ρ) is a continuous function, then f is uniformly continuous. Proof. Suppose f is not uniformly continuous.

It has a convergent subsequence {x2n : n ∈ N2 } with x2 = limn∈N2 x2n . Continue, and we get Nq ⊆ · · · ⊆ N1 ⊆ N and xk = limn∈Nk xkn for 1 ≤ k ≤ q. It follows that {xn : n ∈ Nq } is a subsequence of the original sequence and it converges to x = (x1 , . . , xq ); it must be that x ∈ K since K is closed. 5, K is compact. 9. For the metric space Q, if a, b ∈ Q, a < b, then the set F = {x ∈ Q : a ≤ x ≤ b} = Q ∩ [a, b] is closed and bounded but not compact. In fact, there is a sequence {xn } in F that converges to an irrational number, and so this sequence has no subsequence that converges to a point of F .

2) that for n, m ≥ N , d(xn , xm ) < 2N −1 , and so {xn } is a Cauchy sequence. Since X is complete, there is an x in X such that xn → x. 2) reveals that when n > m > N , x ∈ cl B(xn ; rn ) ⊆ B(xm ; rm ) ⊆ G ∩ UN . ∞ Since N was arbitrary, x ∈ G ∩ N =1 UN . Thus, this last set is nonempty, and so we have established density. The next result is the form of the theorem that is used very often and is also referred to as the Baire Category Theorem. 3. If (X, d) is a complete metric space and {Fn } is a sequence ∞ of closed subsets such that X = n=1 Fn , then there is an n such that we have int Fn = ∅.

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Elementary Concepts In Topology by Alexandroff P.


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