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Math Help - topology/metric spaces questions

  1. #1
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    topology/metric spaces questions

    Hi everyone, I'm studying for a topology exam. Any help would be really appreciated.

    1. (a) Let X be the interval (0, 1/3) with usual Euclidean metric. Show that f : X — X defined by f(x) = x^2
    is a contraction, but f does not have a fixed point in X. Why does this not contradict the Banach
    fixed point theorem?
    (b) Let (X, d) be a complete metric space and f : X — X. Define g(x) =f(f(x)), that is, g = f o f. Assume that the map g : X — > X is a contraction. Prove that f has a unique fixed point.

    2. (a) Let (X, d) be a metric space and let {fn} be a sequence of continuous functions, fn : X — > R, for n € N. Prove that if {fn} converges uniformly to f : X — > R, then f is a continuous function.
    (b) Let fn(x) = (1-xⁿ)/(1+xⁿ) for x € [0, 1] and n € N. Find the pointwise limit f of the sequence {fn}, and determine whether the sequence is uniformly convergent on the interval [0, 1].


    3. (a) Let X be a topological space, and let A and B be compact subsets of X. Prove that the union A U B is a compact subset of X.
    (b) Let f : X —> Y be a continuous map between topological spaces. Prove that if X is compact, then f(X) is compact.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by laura_d View Post
    Hi everyone, I'm studying for a topology exam. Any help would be really appreciated.

    1. (a) Let X be the interval (0, 1/3) with usual Euclidean metric. Show that f : X — X defined by f(x) = x^2
    is a contraction, but f does not have a fixed point in X. Why does this not contradict the Banach
    fixed point theorem?
    d(f(x),f(y))=d(x^2,y^2)=\sqrt{x^4+y^4}

    on (0,1/3) we have x^4\le (1/9) x^2, so:

     <br />
d(f(x),f(y))=\sqrt{x^4+y^4} \le (1/3)\sqrt{x^2+y^2}=(1/3)d(x,y)<br />

    So f is a contraction on (0,1/3).

    There is no fixed point as that would require that:

    x=x^2

    have a root in (0,1/3) and it does not.

    This does not contradict the fixed point theorem as (0,1/3) is not closed, complete or whatever the equivalent condition in the version of the theorem you are using is.

    CB
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