Separated topology and existence of a metric

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Discussion Overview

The discussion revolves around the relationship between separated topological spaces and the existence of a metric. Participants explore whether every separated topological space can be metrized, touching on the conditions and restrictions that may apply.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if it can be proven that every separated topological space has a corresponding metric.
  • Another participant counters that there is a known theorem regarding the conditions under which a space is metrizable, suggesting further research on the topic.
  • Some participants indicate that restrictions on cardinality may be necessary for the discussion, particularly in relation to the definition of separated spaces.
  • A clarification is provided regarding the term "separated," specifically referring to the existence of disjoint open sets for any two distinct points in the space.
  • One participant mentions the need for second countability in compact spaces to ensure metrizability, citing examples where large cardinal products may not be metrizable.
  • Reference is made to Uhryson's lemma as potentially relevant to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on whether every separated topological space can be metrized, with some suggesting that specific conditions must be met, indicating that the discussion remains unresolved.

Contextual Notes

Participants note the importance of cardinality restrictions and the definitions used in the discussion, which may affect the conclusions drawn about metrizability.

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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