Characterizing compactness by properties of continuous functions defined on a space I want to find theorems of the form "all continuous functions defined on a topological space X satisfy some properties if and only if $X$ is compact."
Some restrictions on the space X may be acceptible, such as Hausdorff, metric space, etc. The codomain of the functions could be anything.

From the comments, I found that a metric space is compact if and only if every continuous real function on it is bounded. I would appreciate any other similar theorems.
 A: There is no such characterisation that I know of, but there are some known things:
A space $X$ is called pseudocompact iff every continuous $f:X \rightarrow \mathbb{R}$ is bounded.
For normal spaces ($T_1$ plus being able to separate disjoint closed sets) this is equivalent to countable compactness (every countable open cover has a finite subcover) and limit point compactness (every infinite subset has a limit point).
For metric spaces all three are equivalent to compactness and sequential compactness as well (every sequence has a convergent subsequence).
But not in general, as $\omega_1$, the first uncountable ordinal in the order topology or $\Sigma$-Products of uncountably copies of $[0,1]$ show.
These are normal, pseudocompact (and so countably compact etc.) but not compact.
Using mappings of a special kind : $X$ is compact iff for every space $Y$, the projection $\pi_Y: X \times Y \rightarrow Y$ is a closed map.
If we already knwow that $X$ is $T_3$, $X$ is compact iff every embedding $e$ of $X$ into a Hausdorf space $Y$ has a closed image $e[X] \subset Y$. (in general such a space is called H-closed, and H-closed $T_3$ spaces are compact).
