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Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence

I am having a hard time understanding what the topology of uniform convergence means. I of course know what uniform convergence means - a sequence of functions satisfying a particular set of convergence properties - but the "topology of uniform convergence"? Not sure.

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I would assume it means to view $C(X,\mathbb R)$ as a metric space with the uniform metric $$d(f,g)=\sup_{x\in X}\;|f(x)-g(x)|$$ and derive a topology from that metric. Then convergence of a sequence under this toplogy is the same as uniform convergence of functions $X\to\mathbb R$.

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  • $\begingroup$ But the "topology of uniform convergence" seems to mean something different. Does it mean that the same word is used to mean two different things or does it mean that the "topology of uniform convergence" coincides with the topology induced by the sup-norm in certain cases? $\endgroup$ – Rudy the Reindeer Jul 14 '14 at 11:05
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    $\begingroup$ I get it. Whoever wrote the Wikipedia article misuses "topology of uniform convergence" to mean "topology induced by $\|\cdot\|_\infty$". How terrible. This individual should be banned from writing about mathematics. $\endgroup$ – Rudy the Reindeer Jul 14 '14 at 11:09
  • $\begingroup$ The corresponding wiki seems have been changed, though I was lost reading it. Are there other concerns than the continuous functions may not be bounded? Do they coincide when we consider $C_c (X)$ instead? $\endgroup$ – Syl.Qiu Jan 6 '17 at 7:29

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