Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for more categorical formulation see http://en.wikipedia.org/wiki/Final_topology).

*Does there exist a finite subfamily with the same property?*

Compactness assumption is necessary here, because any open cover of $X$ gives rise to a family of inclusions, for which the topology on $X$ is final. So, the question above can be reformulated as follows: *is it true that a topological space $X$ is compact iff in any family $f_i:Y_i\to X$ which generates the topology on $X$ as the final topology one can find a finite subfamily with the same property.*

**Thank you for fast answers!** I am very impressed. The negative answer means that open coverings are very far from families of maps defining topology. Perhaps, open covering are maps **from** rather than **to** the space, like locally finite coverings. My real motivation was to understand compactness in terms of the category of topological spaces. I cannot shape this as a well-posed problem, but nonetheless still hope for a reasonable answer.