# Explicit choice functions for finite sets in topological spaces

When dealing with finite nonempty sets of real or natural numbers it is always possible to define a explicit choice function, that choose one (arbitrary, but well defined) element out of that set: Take for example the smallest element; or the greatest.

My question is: If we are in a very much more abstract setting, like a (arbitrary) topological space $X$, is then there a way to explicitely define a choice function as above (i.e. a function that for every finite nonempty set of elements in this space returns one element of this sets) ?

I'm aware that this may be a somewhat open question, since the probable answer which awaits me is, I think "there isn't any known explicit choice function" - but that doesn't mean it's proven that no such function exists. The use of the axiom of choice isn't allowed! Also answers that depend on the set-theoretical construction of $X$ don't count, like always taking the element that has least amount of $\emptyset$'s in it - this is just an (incorrect - as we may consider different elements with the same amount of $\emptyset$'s "in" them) example of what I want to avoid. Side question: Would such a function, that depends on the set-theoretic construction of $X$, even be define in the ZFC that I'm working in, can I only define it in metamathematics ?

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There is not: it’s consistent that there be a space $X$ such that no choice function for the non-empty finite subsets of $X$ exists. –  Brian M. Scott Dec 21 '12 at 15:35