Are functions from a non metrizable general topological space into $\mathbb{R}$ continous under the ring structure?

Question

That is, would continous functions from a general topological space X be closed under field addition, multiplication and so on?

Supposing $X$ is metrizable the proof is pretty doable, and example of it being closed under multiplication is shown here : Proof verification: the product of two continuous functions is continuous.

However, how do we prove continuity when it's from a general topological space?

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Here is what I got so far, we have $f,g: X \to Y$ is continous and must prove that $fg$ and $f+g$ is continous.

Since the sum seems simpler I start with that,

Let's suppose try to prove that $f+g$ is continous, we must show that $\text{preim}_{f+g} ( B_{\epsilon} (y) ) = U_x$ where $U_x$ is an open set in $X$ given that $\text{preim}_{f} ( B_{\epsilon} (y) ) = U_1$ and $\text{preim}_{g} ( B_{\epsilon} (y) ) = U_2$

Now this seems quite intractable. Could someone advise on how to proceed further?

Answer 1

Alright, from the hints I got what to do. For those with the same question, here is the solution:

Consider the following commutative diagram of topological spaces:

[]

where,

1. $i: x \to (x,x)$

2. $p: (a,b) \to (f(a),g(b) )$

3. $q: (a,b) \to ab$

The goal is to prove continuity of $ q \circ p \circ i $. We do it in steps, but firstly, a lemma.

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The universal property of product topologies

If all objects in the above diagram are topological spaces, then it simply states that $f_1: Y \to X_1$ and $f_2: Y \to X_2$ being continous implies existence of a unique and conitnous $f$.

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1. i

Consider the constant function $f: X \to X, x \mapsto x$. This is continous in any topology. From the diagram,

we get that the function $f(x,x)=(x,x)$ must be continous.

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2. $p$

We have the following diagram:

Where $ u: (a,b) \mapsto f(a)$ and $v: (a,b) \mapsto g(b)$. We can see both them as composite $u= f( \pi_1 \big( (a,b) \big) )$ and $v= g( \pi_2 \big( (a,b) \big))$, where $ \pi_1 : X \times X \to X, (a,b) \mapsto a$ and $ \pi_1 : X \times X \to X, (a,b) \mapsto b$. Since projections are continous, and we were given that $f,g$ are continous composition are continous.

Hence, $u,v$ is continous which then leads to $p$ being continous by the universal property.

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3. $q$

Well.. this one is obvious. It would also be continous if it were an addition.

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Finally, the entire composition is continous.