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Math Help - Metric Spaces/Topology help

  1. #1
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    Metric Spaces/Topology help

    Any help with these would be hugely appreciated

    (1) Let (X, d) and (Y, d'
    ) be metric spaces, and let f : X Y be continuous with f(X) = Y. Show that if (X,d) is complete and
    d(x,y) < kd'(f(x),f(y)) for some constant k and all x,y X, then (Y, d') is complete.

    (2) Let A be a non-empty compact subset of X. Prove that there exist points a,b A such that d(a, b) = sup{d(x,y) : x,y A}.
    (3)(a) Prove that a topological space X is connected if and only if every continuous map f : X {0,1} is constant, where {0,1} is equipped with the discrete topology.
    (b) Let S
    1 be the unit circle {(x,y) R2 : x2 + y2 = 1} in R2 with the topology induced by the standard topology of R2 . Assume that
    f : S1 R is a continuous map. Prove that there exists a point
    z = (x,y) S1 such that f(z) = f(-z). (Hint: Consider the function
    g(z) = f(z) - f(-z)).

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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by laura_d View Post
    Any help with these would be hugely appreciated







    (2) Let A be a non-empty compact subset of X. Prove that there exist points a,b A such that d(a, b) = sup{d(x,y) : x,y A}.

    Hint: d(a,b)=\text{diam}\left(A\right), consider some Cauchy Sequence \left\{p_n\right\} on A, now use the fact that if \left\{p_n\right\} is a Cauchy Sequence that \text{diam}\left(A\right) exists
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  3. #3
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    Quote Originally Posted by laura_d View Post
    Any help with these would be hugely appreciated

    (1) Let (X, d) and (Y, d') be metric spaces, and let f : X Y be continuous with f(X) = Y. Show that if (X,d) is complete and
    d(x,y) < kd'(f(x),f(y)) for some constant k and all x,y X, then (Y, d') is complete.

    (2) Let A be a non-empty compact subset of X. Prove that there exist points a,b A such that d(a, b) = sup{d(x,y) : x,y A}.
    (1) You need to show that every Cauchy sequence in Y converges. You are told that f is surjective, so a Cauchy sequence in Y must be of the form \{f(x_n)\} for some sequence \{x_n\} in X. Use the information in the question to deduce that \{x_n\} converges to some point z\in X, and conclude that \{f(x_n)\} converges to f(z).

    (2) Let D = \sup\{d(x,y):x,y\in A\}. By the definition of supremum, there are sequences \{x_n\},\ \{y_n\} in A such that d(x_n,y_n)\to D as n\to\infty. Since A is compact, \{x_n\} has a convergent subsequence. Replacing \{x_n\} by this subsequence, you can assume that \{x_n\} converges, to a say. Then do the same for the other sequence \{y_n\}, so that this also converges, to b say. Then conclude that d(x_n,y_n)\to d(a,b).
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