What is the mathematical word to describe when two objects can be transformed to each other like Klein bottle...
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In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?
general-topology terminology
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
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add a comment |
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In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?
general-topology terminology
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
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2
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Homeomorphic?
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– DreamConspiracy
Dec 7 '18 at 16:46
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@DreamConspiracy yes, thank you.
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– katerine
Dec 7 '18 at 16:48
3
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@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
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– Randall
Dec 7 '18 at 16:51
add a comment |
$begingroup$
In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?
general-topology terminology
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
$endgroup$
In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?
general-topology terminology
general-topology terminology
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
edited Dec 7 '18 at 17:19
Brahadeesh
6,50942363
6,50942363
asked Dec 7 '18 at 16:43
katerinekaterine
326
asked Dec 7 '18 at 16:43
katerinekaterine
326
asked Dec 7 '18 at 16:43
katerinekaterine
326
326
2
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Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46
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@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48
3
$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51
add a comment |
2
$begingroup$
Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46
$begingroup$
@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48
3
$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51
2
2
$begingroup$
Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46
$begingroup$
Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46
$begingroup$
@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48
$begingroup$
@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48
3
3
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@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51
$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51
add a comment |
2 Answers
2
active
oldest
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There are a few notions depending on what type of equivalence you want to consider.
• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.
• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.
• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.
Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.
There are also other notions of equivalence like isometry, etc.
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
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add a comment |
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Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
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2
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Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
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– Noah Schweber
Dec 7 '18 at 17:19
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@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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oldest
votes
$begingroup$
There are a few notions depending on what type of equivalence you want to consider.
• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.
• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.
• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.
Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.
There are also other notions of equivalence like isometry, etc.
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
$endgroup$
add a comment |
$begingroup$
There are a few notions depending on what type of equivalence you want to consider.
• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.
• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.
• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.
Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.
There are also other notions of equivalence like isometry, etc.
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
$endgroup$
add a comment |
$begingroup$
There are a few notions depending on what type of equivalence you want to consider.
• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.
• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.
• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.
Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.
There are also other notions of equivalence like isometry, etc.
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
$endgroup$
There are a few notions depending on what type of equivalence you want to consider.
• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.
• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.
• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.
Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.
There are also other notions of equivalence like isometry, etc.
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.5k41741
10.5k41741
add a comment |
add a comment |
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Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
$endgroup$
2
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
add a comment |
$begingroup$
Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
$endgroup$
2
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
add a comment |
$begingroup$
Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
$endgroup$
Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
edited Dec 7 '18 at 17:21
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9391216
9391216
2
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
add a comment |
2
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
2
2
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
$endgroup$
– Noah Schweber
Dec 7 '18 at 17:19
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
$begingroup$
@NoahSchweber brain fart, thank you
$endgroup$
– DreamConspiracy
Dec 7 '18 at 17:22
add a comment |
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2
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Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46
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@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48
3
$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51