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If X is compact then so is C[X].(C[X] is the set of all continuous functions over X.) Does there exit a Necessary and Sufficient Condition here ;does compactness of C[X] say anything about X?

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  • Is $C[X]$ the space of real- (or complex-) valued functions on $X$, or of $X$-valued functions? – Daniel Fischer Oct 10 '14 at 12:18
  • real /complex valued sir – Learnmore Oct 10 '14 at 12:21
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    Then $C[X]$ is never compact unless $X = \varnothing$. It is then always a (real or complex) vector space, with a Hausdorff topology. Such a vector space is compact if and only if it is the trivial vector space. – Daniel Fischer Oct 10 '14 at 12:23
  • @learningmaths Under what topology on $C[X]$? If you mean under the topology induced by the supremum norm, the statement is false. – egreg Oct 10 '14 at 12:23

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