$\dim$ $C(X,\mathbb{R})<\infty$ we need to show $|X|<\infty$,

Question

$X$ be a compact hausdorf space such that $\dim$ $C(X,\mathbb{R})<\infty$ we need to show $|X|<\infty$, I must say I have no idea how to prove this result. please help. Thank you!

Answer 1

Let $N$ be the dimension of $C(X,\Bbb R)$. Assume that there are $N+1$ distinct points $x_1,\dots,x_{N+1}$ in $X$. As $X$ is Hausdorff, by Urysohn lemma, we can find for each $j$ a continuous function $g_j\colon X\to\Bbb R$ such that $g_j(x_k)=0$ when $k\neq j$ and $g_j(x_j)=1$.

The family $\{g_j,1\leqslant j\leqslant N+1\}$ is necessarily linearly dependent, so we can assume that $g_{N+1}=\sum_{j=1}^Ng_j$. Evaluating this equality at $x_{N+1}$ yields a contradiction.

This proves that $\dim C(X,\Bbb R)=\operatorname{card}(X)$.

The result is not necessarily true when $X$ is not assumed to be Hausdorff. For example, take $X$ an infinite set with the topology $\{\emptyset,X\}$. The only continuous functions are constant ones.