Can we proove that for any separated topological space, there exists a metric?
No, I don't think so. THere is a well known theorem that states when a space is metrizable. Try googling for it.
Appears you must have some kind of restriction on the cardinality of some things.
(exactly what do you mean by separated?)
sorry: direct french translation.
for any two different points (x,y) of this set, I have at least two disjoint open sets (A,B), such that x element of A and y element of B.
And yes, I think this is theorem I have forgotten about metrizable spaces, I am searching it now again.
you need second coutable (if it is compact) so something like a product of [0,1] indexed by some very large cardinal won't be metrizable.
see also Uhyrson's lemma
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