Are pointwise addition,multiplication continuous on the space of continuous real functions, with the product topology?

Question

Let $A$ be a topological space. Let $F$ be the set of functions $A \rightarrow \mathbb{R}$ which are continuous. Give $F$ the operations of pointwise addition and multiplication. Thus $F$ becomes a (unital, comm, assoc) real algebra.

I was wondering, in general are the above operations on $F$ continuous (as maps $F\times F \rightarrow F$) if we give $F$ the subspace topology from $\mathbb{R}^A \supset F$, where $\mathbb{R}^A$ has the product topology?

I'm most interested in the cases where $A = \mathbb{N},\mathbb{Z}$ or $A = \mathbb{R}$. Currently I'm trying to work through the details, but I would greatly appreciate if anyone could give any suggestions or suggest any references/sources that discuss the above.

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Edit: An earlier version of this question asked about the case when $F$ has the compact-open topology. However, according to Wikipedia, composition is continuous for the compact-open topology if the "middle" space is LCH; so the above operations should be continuous if $F$ is given the compact-open topology. Hence, I have edited the question to focus on the case of the product topology.

Edit: changed the question to allow $A$ to be any topological space.

Answer 1

• I assume you consider the standard distance on $\Bbb R$ (otherwise addition can be discontinuous).
• If I understand correctly, $F= C(A,\Bbb R)\subset \Bbb R^A$.
• Let $\star\colon \Bbb R^2\to \Bbb R$ be an addition or multiplication or any other continuous function.
• In what follows by $\{f,g\}\colon A\to \Bbb R\times \Bbb R$ for $f,g\in \Bbb R^A$ I denote something that is commonly denoted by $(f,g)$ and is given by $\{f,g\}(a)=(f(a),g(a))$. The change of the symbol is in order to avoid ambiguity with the pair $(f,g)\in F^2$.
• Moreover let $f\times g$ be given by the formula $(f\times g)(a,b)=(f(a),g(b))$.
• It's well known that if $f,g$ are continuous then $\{f,g\},f\times g$ are also continuous.
• Let $\phi\colon F\times F\to F$ be given by the formula $\phi(f,g)(a)=f(a)\star g(a)$. Observe it's well defined since $\phi(f,g) = \star \circ \{f,g\}$ is continuous as a composition of two continuous functions.
• To show that $\phi$ is continuous, it sufficies to show that the compostion $i\circ \phi$ is continuous where $i\colon F\subset \Bbb R^A$. From the definition of the product topology we have to show that for any $b\in A$ the composition $\psi_b=\pi_{b}\circ i\circ \phi$ is continuous, where $\pi_b\colon \Bbb R^A\to \Bbb R$, $\pi_b(f)=f(b)$ is a projection.
• Observe that $$\psi_b(f,g)=f(b)\star g(b) = (\star\circ (\pi_b\times\pi_b))(f,g),$$ so $\psi_b= \star\circ (\pi_b\times\pi_b)\circ (i\times i)$ is continuous, as all the projections are continuos.