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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.2012-12-21

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It is consistent that there are sets which can be expressed as a countable union of pairs, but there is no choice function from the pairs.

Given such set, how could you expect to find a choice function on its finite subsets? Regardless to what the topology is, that would be impossible.

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    @temo: Yes, but this rule depends on a parameter. For example what sort of rule would you propose for choosing from finite subsets of $\mathcal P(\mathbb R)$? Without the axiom of choice it is consistent that there is no way to describe this rule, even using intangible sets as parameters; and with the axiom of choice we can make it so no choice function can actually be given if one requires it to have no parameters. In this sense, the real numbers are the pathological set because they allow us to make such choice explicitly.2012-12-22