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nzur
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Join Date: Apr 2008
Posts: 96
Good Answers: 1

Topological Equivalence

03/29/2018 2:46 AM

Hi folks,

A bijective map establishes a one-to-one correspondence between elements and has an inverse. If a map is homeomorphism (bijective and has a continuous inverse), the map garantees topologically equivalent transformation.

In topological view point, a donut and a cup are equivalent structures.

I just want to (intuitively) understand how the bijective map garantees topological equivalence in topological structure mapping. How do I know transformed structure is topologically equivalent after the one-to-one mapping with inverse condition.

Thanks!
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SolarEagle
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Join Date: Mar 2007
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#1

Re: Topological equivalence

03/29/2018 2:59 AM

It must meet certain criteria...

http://www.math.wsu.edu/math/faculty/schumaker/Math512/512F10Ch4B.pdf

https://math.stackexchange.com/questions/165434/how-to-prove-if-a-function-is-bijective

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Randall
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Join Date: Aug 2005
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#2

Re: Topological equivalence

03/29/2018 6:22 AM

I just want to (intuitively) understand how the bijective map garantees topological equivalence in topological structure mapping.

I don't think it does.

Take a lump of clay formed into a doughnut and number all the atoms; squish it into a ball; the numbering automatically defines the bijective mapping, but, the two lumps are not topologically equivalent.

OK: the atoms are countable but I'm pretty sure you could relatively easily define a mapping which continuously mapped every point in the vicinity of atoms n and n+1 in the ball to every point in the vicinity of the same two atoms in the doughnut.

Or maybe if those two atoms have now been separated by the "hole" you get a discontinuity where the mapping breaks down.

So I think I've changed my mind, and, following this reasoning might help you?

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PWSlack
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United Kingdom - Member - Indeterminate Engineering Fields - Control Engineering - New Member

Join Date: Jan 2007
Location: In the bothy, 7 chains down the line from Dodman's Lane level crossing, in the nation formerly known as Great Britain. Kettle's on.
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#3

Re: Topological Equivalence

03/29/2018 9:03 AM

https://cr4.globalspec.com/comment/1242642

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Deefburger
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Join Date: Mar 2011
Location: Sebastopol, California
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#4

Re: Topological Equivalence

03/30/2018 10:29 AM

"In topological view point, a donut and a cup are equivalent structures."

No they are not, unless the cup has a handle and therefore a hole.

https://www.youtube.com/watch?v=k8Rxep2Mkp8

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