product of two continuous functions on topological space X is or not continuous?

Question

product(pointwise) of two continuous functions on $X =\mathbb{R}^n$ is still continous because we have $\lim_{x\to x_0} f*g(x)= \lim_{x\to x_0} f(x) \cdot \lim_{x\to x_0}g(x)$.

In a more general case, suppose $X$ is a topological space, is it still holds?

Answer 1

If $f,g:X \to \mathbb R$ are continuous then $(f,g): X \to \mathbb R^{2}$ is continuous and the map $(x,y) \to xy$ is a continuous map from $\mathbb R^{2}$ to $\mathbb R$. Hence their composition $fg $ is continuous.