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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?

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Is this a homework problem? – Clive Newstead Sep 19 '12 at 18:50
$2$ 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? – Chris Eagle Sep 19 '12 at 19:23

Here's a hint for #1: Let $\Gamma(f) = \{ (x, f(x)) : x \in X \}$ (called the graph of $f$). Consider the two projections $\pi_1 : \Gamma(f) \rightarrow X$ and $\pi_2 : \Gamma(f) \rightarrow \mathbb{R}$. Given a closed set $K \subset \mathbb{R}$, show that $$f^{-1}(K) = \pi_1(\pi_2^{-1}(K)).$$ Use this decomposition to prove that $f^{-1}(K)$ is compact and thus closed.

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Let $\Gamma(f) = \{ (x, f(x)) : x \in X \}$ be the graph of $f$. Assume $f$ is not continuous. Then there is a point $x_0$ in $X$ and there is an open neighbourhood $V$ about $f(x_0)$ whose preimage $f^{-1}(V)$ is not a neighbourhood of $x_0$. For each point $x\neq x_0$ there are disjoint open neighbourhoods $U_x$ and $U^0_x$ about $x$ and $x_0$ respectivly.
Now show that $\{U_x\times\mathbb{R}:x\neq x_0\}\cup\{X\times V\}$ is an open cover of $\Gamma(f)$ without finite subcover.

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HINT for (2): Assume that $X$ is infinite. First show that there is a family $\{U_n:n\in\Bbb N\}$ of pairwise disjoint, non-empty open sets. One way to do this is to pick a point $p\in X$ that is not an isolated point, and let $V_0$ be any open nbhd of $p$. Pick $x_0\in V_0\setminus\{p\}$, and let $V_1$ be an open nbhd of $p$ such that $\operatorname{cl}_X V_1\subseteq V_0\setminus\{x_0\}$. Continue in the same fashion: given $V_n$, pick $x_n\in V_n\setminus\{p\}$, and let $V_{n+1}$ be an open nbhd of $p$ such that $\operatorname{cl}_X V_{n+1}\subseteq V_n\setminus\{x_n\}$. For $n\in\Bbb N$ let $U_n=V_n\setminus\operatorname{cl}_X V_{n+1}$; I’ll leave it to you to verify that the sets $U_n$ are pairwise disjoint and non-empty.

Now for each $n\in\Bbb N$ let $f_n:X\to[0,1]$ be a continuous function such that $f_n(x_n)=1$ and $f_n(x)=0$ for all $x\in X\setminus U_n$. (How do I know that there is such a function?) Now show that the functions $f_n$ are linearly independent.

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