Does every Hausdorff space admit a metric?

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Discussion Overview

The discussion centers on the relationship between Hausdorff spaces and metrizable spaces, specifically whether every Hausdorff space admits a metric. Participants explore the conditions required for metrizability, including separability and the existence of a countable locally finite cover, and question the necessity of paracompactness in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that every metric space is Hausdorff, but not every Hausdorff space is metrizable.
  • One participant cites a source claiming that metrizability requires Hausdorffness, separability, and a countable locally finite cover, noting that these conditions are independent.
  • Another participant expresses uncertainty about the converse of the statement regarding metric spaces and Hausdorff spaces.
  • Several participants seek proofs or references that establish the conditions necessary for metrizability.
  • One participant mentions that paracompactness might also be a criterion for metrizability, prompting discussion about its relationship to countable locally finite covers.
  • A later reply clarifies that paracompactness implies the existence of a countable locally finite refinement for open covers, suggesting a nuanced understanding of the criteria for metrizability.

Areas of Agreement / Disagreement

Participants generally agree that there are specific conditions for metrizability, but there is no consensus on whether paracompactness is necessary or if it can be excluded. The discussion remains unresolved regarding the exact criteria required for a space to be metrizable.

Contextual Notes

The discussion highlights the complexity of the relationships between different topological properties and the conditions for metrizability, indicating that assumptions about these properties may vary among participants.

Stevo
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Every metric space is Hausdorff but not every Hausdorff space is metrizable!

Googling on "Hausdorff" and "metrizable", I found
"Metrizable requires, in addition to Hausdorf, separability and existence of at least one countable locally finite cover. Those three are independent requirements; if you could do without anyone of them you would have a much stronger theorem, and be famous among topologists (nobody else would notice or care)." attributed to a "DickT" on

http://superstringtheory.com/forum/geomboard/messages3/143.html

apparently a "string theory" message board.
 
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That was me, and I stand behind it. I should, because I got it straight out of one of my old textbooks!
 
HallsofIvy said:
Every metric space is Hausdorff but not every Hausdorff space is metrizable!

Googling on "Hausdorff" and "metrizable", I found
"Metrizable requires, in addition to Hausdorf, separability and existence of at least one countable locally finite cover. Those three are independent requirements; if you could do without anyone of them you would have a much stronger theorem, and be famous among topologists (nobody else would notice or care)." attributed to a "DickT" on

http://superstringtheory.com/forum/geomboard/messages3/143.html

apparently a "string theory" message board.

Yeah, it's pretty easy to show that every metric space is Hausdorff... I wasn't sure if the converse was true. Thanks for that.

Does anybody have a proof, a link to a proof, or a reference to a proof that metrisation requires Hausdorff, separability, and existence of a countable locally finite cover?
 
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Stevo said:
Does anybody have a proof, a link to a proof, or a reference to a proof that metrisation requires Hausdorff, separability, and existence of a countable locally finite cover?

Try these:-
1) Manifolds at and beyond the limit of metrisability at arXiv:math.GT/9911249
2) http://www.math.auckland.ac.nz/~gauld/research/ (the file is labelled metrisability.pdf)
both by David Gauld at University of Auckland Department of Mathematics.

A mathematical physics prof taught me that paracompactness must also be one of the criteria of metrisability.
Can there really be a proof that doesn't include this criteria?
 
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Paracompactness is a generalization from the countable locally finite cover. If a space is paracompact then every open cover of it has a countable locally finite refinement. So you get a little narrower theorem by specifying the CLF cover specifically, but in many instances, you would use the given paracompactness of the space to prove the CLF cover exists.

The theorem is called Urysohn's theorem. http://www.cs.utk.edu/~mclennan/Classes/594-MNN/MNNH/MNNH-3/node20.html is a sketch of the proof.
 
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