Separated topology and existence of a metric

seratend
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Can we proove that for any separated topological space, there exists a metric?

Seratend.
 
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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?)
 
matt grime said:
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.

Seratend
 
oh, Hausdorff.

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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