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There are two topology problems:

  1. Let $X$ be a Hausdorff space. Let $f : X \to \mathbb{R}$ be such that $\{(x, f(x)) : x \in X\}$ is a compact subset of $X \times \mathbb{R}$. Show that $f$ is continuous.

  2. Let $X$ be a compact Hausdorff space. Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional. Show that $X$ is finite.

Please help, how can I solve these problems?

  • 1
    Is this a homework problem?2012-09-19
  • 0
    $2$ is equivalent to showing that the space of continuous functions is infinite-dimensional when $X$ is infinite. Can you show that there are *any* non-constant real-valued continuous functions on an infinite compact Hausdorff space?2012-09-19

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