Uniform continuity and compactness

Question

We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.

Do we have a generalization of this theorem for general topological space.

Answer 1

To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.

Here's the definition of uniform continuity for metric spaces:

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.

A function $f:X\to Y$ is uniformly continuous if

$$\forall\varepsilon>0,\exists\delta>0,\forall x,y\in X, d_X(x,y)<\delta\implies d_Y(f(x),f(y))<\varepsilon.$$

Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.

Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.

Your result generalizes to this:

Let $X$, $Y$ be uniform spaces and $f:X\to Y$ be a continuous function.

Suppose that $X$ is compact. Then $f$ is uniformly continuous.

I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.