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Let $X$ be a non-empty compact Hausdorff space.Show that if $X$ has at least $n$ distinct points then the dimension of $C(X)$ the space of continuous real valued functions defined on $X$ is at least $n$.

How should I prove this fact?Please give some hints as I am stuck completely

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  • $\begingroup$ Do you need further comments or more details? $\endgroup$ Commented Oct 16, 2016 at 14:14
  • $\begingroup$ No I have got my answer@RenanManeliMezabarba $\endgroup$
    – Learnmore
    Commented Oct 16, 2016 at 14:31

1 Answer 1

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A compact Hausdorff space is normal, hence it is particularly a completely regular space.

Thus, if $|X|\geq n$, fix $F\subset X$ with $|F|=n$ and, for each $x\in F$, let $f_x\colon X\to \mathbb{R}$ be a continuous function such that $f_x(x)=1$ and $f_x(y)=0$ for each $y\in F\setminus\{x\}$.

Notice that $\{f_x:x\in F\}$ is a linear independent subset of $C(X)$.

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