I have a stupid question about a topology on $C\_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.


**Definition (topology on $C\_c(X)$):** For each compact $K \subset X$, $C\_K(X)$ is the set of functions in $C\_c(X)$ with support in $K$. $C\_K(X)$ is given a Banach space structure with the sup norm. Call this topology $\tau\_K$.


Let $\beta$ be the set of $V \subset C\_c(X)$ which are convex, balanced, and which have $V\cap C\_K(X) \in \tau\_K$ for each compact $K$.


Define $\tau$ to be the collection of sets in $C\_c(X)$ given by $\bigcup (\phi\_i + W\_i)$ where $W\_i \in \beta$. This gives a topology on $C\_c(X)$ making it a locally convex topological vector space with $\beta$ as a local basis. Moreover, the topology $\tau\_K$ coincides with the subspace topology $\tau|\_{C\_K(X)}$.


For all the above facts see Rudin's *functional analysis* $\S 6.3$ onwards.


**Question:** Take the product topology $\tau \times \tau$ on $C\_c(X) \times C\_c(X)$. Say $V\subset C\_c(X)\times C\_c(X)$ is convex and balanced.
Suppose for every pair of compact sets $K, F \subset X$, we have that $V \cap C\_K(X)\times C\_F(X) \in \tau\_K \times \tau\_F$. Is $V$ open in the product topology $\tau\times \tau$?


I came across this question while studying pseudomeasures and induced representations from Folland's *Harmonic analysis*.


**Attempt:** Say $(\phi\_1, \phi\_2) \in V$. It suffices to find $W\_1, W\_2 \in \beta$ such that $(\phi\_1,\phi\_2) + W\_1\times W\_2 \subset V$...Pretty lost at this point.


Also posted [here](https://math.stackexchange.com/questions/4326256/product-of-inductive-limit-topologies-on-c-cx-times-c-cx).



Product of inductive limit topologies on $C_c(X)\times C_c(X)$
The topology $\tau$ you describe makes $(C\_c(X),\tau)$ the *colimit (or inductive or direct limit) in the category LCS of locally convex spaces* of the system $C\_K(X)$ with inclusions $i\_{K,L}:C\_K(X) \hookrightarrow C\_L(X)$ for compact subsets $K\subseteq L$ of $X$, i.e., the inclusions $i\_K:C\_K(X)\to C\_c(X)$ have the *universal property* that, for every family of continuous linear maps (the morphisms of the category) $f\_K:C\_K(X)\to Y$ with $f\_L\circ i\_{K,L}=f\_K$, there is a unique morphism $f:C\_c(X)\to Y$ with $f\_K=f\circ i\_K$. A big advantage of the categorical viewpoint is that you easily get permanence properties like *colimits commute with colimits*.


Since the category LCS is *additive* (you can add two morphisms $(f+g)(x)=f(x)+g(x)$ and this distributes over the composition -- sometimes the term *pre-additive category* is used) general abstract nonsense tells you that the product is also a coproduct and hence a colimit which commutes with colimits.


Therefore and without any calculations, the product topology $\tau\times \tau$ is the colimit topology of the system $C\_K(X)\times C\_L(X)$.


**EDIT.** I hope that the following helps. Whenever you have colimits $E=$colim$\_I E\_i$ and $F=$colim$\_J F\_j$ in a category having codropucts (denoted by $\oplus$) it is always true that $E\oplus F =$colim$\_{I\times J} E\_i\oplus F\_j$. Since LCS is additive, coproducts and products are "the same" so that you can replace $\oplus$ by $\times$. (You see that I am quite reluctant to replace the abstract argument by concrete calculations.)


