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I found the following example in Conway's Functional Analysis Book:

Suppose $X$ is a completely regular space and let $C(X)=$ all continuous functions from $X$ into $\Bbb{C}$. If $K$ is a compact subset of $X$, define $p_K(f)=\sup \{|f(x)|: x \in K\}$. Then $\{p_K: K \text{ compact in }X \}$ is a family of seminorms that makes $C(X)$ into a LCS.

Now for any space $X$ (not necessarily completely regular), $C(X)$ is a vector space. So, once we prove that family $\{p_K: K \text{ compact in }X \}$ of seminorms separates points on $C(X)$ then it would induce a locally convex topology on $C(X)$. Now for $f\in C(X)\setminus \{0\}$ there is $x_0\in X$ so that $f(x_0)\ne 0$. But then $p_{\{x_0\}}(f)=|f(x_0)| \ne 0$. This implies that family of seminorms separates points on $C(X)$ (we didn't use "complete regularity"!) and hence define a locally convex topology on $C(X)$.

My question: Isn't the case that the space $C(X)$ for any topological space $X$, not necessarily completely regular, satisfies the assertion of that example with the same family of seminorms? If so then is there any specific reason for restricting our attention to "completely regular" spaces $X$?? I mean, I know that "completely regular" spaces are "nicer" than general topological spaces, but can anyone mention some important property/result (related to this locally convex topology) on $C(X)$ that wouldn't hold without the "complete regularity" on $X$.

I want to realize the importance of assuming "complete regularity" condition in this case.

Thanks

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It's not necessary but it doesn't hurt: if $(X, \tau)$ is any space at all, we can define a new topology $\tau'$ on the set $X$ so that $Y:=(X,\tau')$ has the same set of continuous functions as $(X, \tau)$ and $(X,\tau')$ is completely regular. And the identity $i:X \to Y$ is continuous and $C(Y) \to C(X)$ defined by $f \to f \circ i$ is a ring isomorphism (and a linear isomorphism too).

So for the study of spaces of the type $C(X)$ (as rings, as is also commonly done, or as linear spaces) it is enough to restrict to completely regular spaces.

Moroever, if $X$ is Tychonoff, there is a more direct connection between properties of the space $X$ and its $C(X)$. We get points from its Cech-Stone compactification $\beta X$ by considering maximal ideals in $C(X)$ as a ring etc. Most spaces that occur in analysis are Tychonoff (all separated TVS, etc) anyway and many $C(X)$ properties are studied for locally compact Hausdorff spaces (also Tychonoff), e.g. the dual via Riesz representation theorems. So the analysis theory is often nicer for such spaces.

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    $\begingroup$ A lot of extra info on the naturalness of completely regular spaces can be found on its Wikipedia page. $\endgroup$ May 17, 2021 at 13:18
  • $\begingroup$ @ Henno Brandsma, How can one define $\tau'$ on $X$ in the first paragraph? $\endgroup$
    – sigma
    May 19, 2021 at 1:54
  • $\begingroup$ @indrajit the so-called weak topology. The minimal topology such that all $\tau$-continuous real-valued functions on are still continuous. $\endgroup$ May 19, 2021 at 4:47
  • $\begingroup$ thank you; I got it now $\endgroup$
    – sigma
    May 19, 2021 at 5:00

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