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Property of lower semi-continuous maps

  1. Feb 15, 2012 #1
    [itex][/itex]1. The problem statement, all variables and given/known data

    If X is a metric space then a map [itex] f: X \rightarrow \mathbb{R} [/itex] is called lower semi-continuous if for each ray of the type [itex] (a, + \infty) [/itex], the inverse image of the ray under f is also open. If X is a compact metric space, prove that f is bounded below, and that f actually obtains its minimum value for some [itex] x \in X [/itex].

    2. Relevant equations



    3. The attempt at a solution

    Here is my proof:

    Let [itex] U_a = (a, + \infty) [/itex]. Let [itex] U = \{U_a : a \in f(X)\} [/itex]. Let [itex] V_a = f^{-1}(U_a) [/itex] and let [itex] V = \{V_a : a \in f(X) \} [/itex]. Then V is an open cover of X and so it reduces to a finite subcovering [itex] \{ V_{a_1}, V_{a_2}, ... , V_{a_n} [/itex]. Let [itex] b = min \{a_1, a_2, ... , a_n [/itex]. Then [itex] f(V_b) = (b, + \infty) \supset (a_i, + \infty) [/itex] for [itex] a_i \neq b [/itex].

    Now we show that f is bounded below by b. Choose any [itex] x \in X [/itex]. [itex] x \in V_{a_i} [/itex] for some [itex] i \in \{1, 2, ... , n \} [/itex]. Thus [itex] f(x) \in (a_i, + \infty) \subset (b, + \infty) [/itex]. Thus f is bounded below. Furthermore, since [itex] b \in U_b \subset f(X) [/itex], there is a c in X such that [itex] f(c) = b [/itex]. Thus f also attains its minimum value of b at the point c of X.

    Is this proof correct?
     
  2. jcsd
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