In the basic analysis, we proved the following with the $\epsilon – \delta$ method.
For metric spaces $X$ and $\mathbb{R}$, $fg:X\rightarrow \mathbb{R}$ is continuous, if $f,g:X\rightarrow \mathbb{R}$ are continuous.
In my text book (Munkres, Topology), I found that the author uses the above property when $X$ is a (normal) space. (For the proof of the Tieteze extension into $\mathbb{R}$; $\forall$ A clsoed subset $A$ of a normal space $X$, a continuous function $f:A\rightarrow \mathbb{R}$ can be continuously extended to $\bar f:X\rightarrow \mathbb{R}$ s.t. $\bar f(x)=f(x)$ $ \forall x \in A$)
Is one able to use the above property for continuous functions from an arbitrary topological space into the real field?
Any help will be appreciated. Thank you.
Best Answer
Yes, this is true more generally when mapping any space into any topological group $G$, since the multiplication map $ \mu : G \times G \to G$ is continuous by definition. Given maps $f, g: X \to G$ their product $f \cdot g : X \to G$ is just the composite of the pairing $(f, g) : X \to G \times G$ with the multiplication. That is, $f \cdot g = \mu \circ (f, g)$.