Is a quotient of continuous functions continuous in any topological space?

Question

I was looking at this question Continuity of rational functions between affine algebraic sets, and in one of the comments it is stated "The fact that a[n] [exp]licit quotient of continuous functions is continuous and that, as you write, a f[u]nction with continuous components is continuous are purely general properties of continuous functions and are are not proper to the Zariski topology."

This comment got me wondering. Suppose I have two continuous functions $f,g: X \rightarrow Y$, $X,Y$ are some topological spaces (say subset of $k^n$ and $k^m$ respectively where $k$ is a field). Suppose that $f/g$ makes is well defined on some open subset $U$. Then is it always continuous on $U$? (and how I can prove that?)

To define $f/g$ You will need some more structure on $Y$, don't You? — Peter Melech, May 31 '19 at 10:28
The answer depends on what you mean by saying that $f/g$ makes sense. — Kavi Rama Murthy, May 31 '19 at 10:28
$g$ is not zero implies You have at least the structure of an abelian group on $Y$, $f/g$ implies You even have a multiplication and the images of $g$ even have inverses, in the first line You say $X,Y$ some topological spaces...could You clarify this? — Peter Melech, May 31 '19 at 10:36
Thanks, a little bit clearer now, $Y\subset k^m$ for $m>1$ still makes not much sense though, e.g. $Y\subset k^{m\times m}$ and $g(X)\subset Gl_m(k)$ would make sense ... — Peter Melech, May 31 '19 at 11:12

score 3 · Accepted Answer · answered May 31 '19 at 11:23

There are many ways to require that topology and algebraic operations are compatible. Usually the requirement is some form of "any algebraic operation is a continuous function", meaning this question is trivial.

But if no such requirement is made, then there is no reason to believe that the algebraic operations are well-behaved with respect to the topology.

Note that if you have specific topological spaces and a specific set of algebraic operations in mind, then you can ask the question of whether they fit together according to some set of criteria. But then you will have to tell us exactly what the spaces, topologies, algebraic operations and criteria are. It's not really a question which can be answered in generality.

Is a quotient of continuous functions continuous in any topological space?

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