Extending a topology to linear combinations? Suppose I have a topological space $X$, and some arbitrary field $K$.  I am trying to nicely describe a set of functions on ${}_K X$, the set of $K$-linear combinations of values in $X$.  I feel like part of the answer I want is continuity, but I don't know how to extend the topology on $X$ to ${}_K X$ nicely.  Are there standard ways to do this?
$K$ should be treated as discrete.  Mild assumptions about $X$ are probably okay.  In particular, $X$ is Hausdorff and compact, if those help.
 A: One thing you could do would be to sort of mimic the ideas of Gelfand duality. However, this will be of most use if $X$ is "zero-dimensional" in the sense that its topology admits a basis of clopen sets.


*

*Let $C(X,K)$ be the vector space (over $K$) of continuous maps $X 
\to K$. Note a function $f \in C(X,K)$ is constant on every connected-component of $X$ since $K$ is discrete.

*There is an mapping $x \to \widehat x : X \to C(X,K)^*$ which sends $x$ to the point evaluation functional given by $\widehat x(f) = f(x)$.  This mapping is typically not injective. If $x_1$ and $x_2$ are in the same component of (or even the same quasi-component, although there is no difference for a compact Hausdorff space) of $X$, then $\widehat x_1 = \widehat x_2$. 

*You can topologize the dual $C(X,K)^*$ with the "weak-star" topology (maybe not the correct terminology to use when dealing with a discrete field) by taking the coarsest topology such that $\varphi \to \varphi(f) : C(X,K)^* \to K$ is continuous for each $f \in C(X,K)$. In particular, a net of point evaluations $\widehat x_i$ will converge to $\widehat x$ if and only if $f(x_i) \to f(x)$ for every $f \in C(x)$. In other words, if and only if, for every $f \in C(X,K)$,  one has $f(x_i) = f(x)$ for large enough indices $i$. In other words, if and only if $x_i$ is eventually in any clopen neighbourhood of $x$. 

*You can topologize the space $\bigoplus_{x \in X} K$ of $K$-linear combinations of points of $X$ with the coarsest topology which makes the unique linear map $\bigoplus_{x \in X} K \to C(X,K)^*$ extending $x \mapsto \widehat x : X \to C(X,K)^*$ continuous. In the case where the latter map is injective (i.e. when there is a basis of clopen sets for $X$), this topology on $\bigoplus_{x \in X} K$ induces the original topology on $X$. Otherwise, I suppose it will induce some weaker topology on $X$ which fails to seperate points of $X$ in the same connected component.

