What is the definition of a bounded function on a topological space?

Question

Can someone please provide a definition of bounded function on a topological space?

I am confused by:

1. What does it mean for a function to be bounded. I know what it means for a set to be bounded i.e. $\text{diam}(X) < \infty$

2. What does it mean for a function to be bounded on a topological space. Because boundedness requires metric, so I don't get why it would make sense.

Context: I am trying to prove that given $(X, \mathfrak{T})$ a compact topological space, $f: X \to \mathbb{R}$ continuous, then $f$ is bounded.

Answer 1

A function $f:X\to\mathbb{R}$ is bounded if the set $f(X)$ is a bounded subset of $\mathbb{R}$.

Equivalently, $f$ is bounded if there exists $M>0$ such that $|f(x)|\leq M$ for all $x\in X$.