Talk:Homeomorphism

From Wikipedia, the free encyclopedia

This is the talk page for discussing improvements to the Homeomorphism article.
Article policies
WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class High Priority  Field: Topology

Contents

[edit] Knot example is not an example

The trefoil knot being deformed into the torus in 4-space is not an example of a homeomorphism, such a deformation is referred to as an ambient isotopy. Any torus embedded in 3-space so as to appear knotted is still a torus, and so is indeed "homeomorphic to a torus" in some tautological sense of the word.

Being knotted is only a property of the embedding, and not a property of the object itself. See, e.g. Knots and Links by Dale Rolfsen. 137.82.36.10 01:57, 19 September 2007 (UTC)

[edit] Correct spelling?

Isn't the correct spelling homoeomorphism rather than homeomorphism? --fil

Not in everyday use.

Charles Matthews 15:20, 9 Feb 2004 (UTC)

Ok I don't use this term every day, and I also thought homeomorphic is the correct spelling; but my spell checker (ispell-emacs) suggests homoeomorphic instead of homeomorphic. Maybe one should point out the alternative spelling. --fil

On the whole, I'd stick with the way humans spell it.

Charles Matthews 16:28, 9 Feb 2004 (UTC)

(Sorry to re-open an old thread.) Some people also leave off the first "o" in oedema and oesophagus. Might this be a English vs American thing? (People outside the US are humans too...) Alternatively, I notice that "homoeomorphism" is defined by several online dictinaries along the lines of "A near similarity of crystalline forms between unlike chemical compounds". If there are indeed two distinct words, that would explain the spell checker's response. LachlanA (talk) 02:49, 7 May 2008 (UTC)


homeomorphism makes perfect sense when speaking only of metric spaces, with no reference to topology. The article needs to be edited. I'm not sure how to word the definition to reflect this. --HellFire 13:43, 18 July 2006 (UTC)

Yeah, but any metric space is automatically a topological space. Oleg Alexandrov (talk) 15:30, 18 July 2006 (UTC)
oh thts fine in tht case.know a little about metric spaces, dont know anything about topology :-) --HellFire 10:56, 19 July 2006 (UTC)

[edit] Animation

The animation is good example of homotopy not of homeomorphism. I suggest it to be placed there, and find another image for exemplifying homeomorphism. SurDin 12:50, 7 January 2007 (UTC)

I changed the caption a bit to make it clear that homeomorphic != homotopy. Maybe we need a counter example of a homeomorphism which is not a homotopy, perhaps two linked ring and two disjoint rings? --Salix alba (talk) 22:20, 12 March 2007 (UTC)

[edit] Confusion

So two objects are homeomorphic if a one-to-one, continuous,invertible function (as opposed to a continuous deformation) mapping one to the other exists. Does this mean that turning an object inside out is an acceptable homeomorphism? What exactly are the restrictions on cutting and gluing to preserve topological properties (it seems a cut+glue pair, done correctly is acceptable)? Thanks.

It seems that only if orientability is preserved. Take for example a cylinder S^1\times I. Fix a point z in the circle and then cut the cylinder along J=z\times I. If you paste along J by identity you re-get the cylinder, but if you paste using a reflexion thru z\times \frac{1}{2}, you will get the mobius strip which is not homeomorphic to the cylinder.--kiddo 17:08, 8 April 2007 (UTC)
This is almost off-topic, but I think the sentence "Homeomorphisms are the isomorphisms in the category of topological spaces" from the NOTES section should be moved to the top of the page. That was the most informative and (for anyone who's taken high-school algebra) clear description in the entire article. —Preceding unsigned comment added by 71.198.178.194 (talk) 18:50, 11 May 2008 (UTC)