What is the mathematical word to describe when two objects can be transformed to each other like Klein bottle...
$begingroup$
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
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
$endgroup$
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
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
$endgroup$
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
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
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
$endgroup$
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
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
edited Dec 7 '18 at 17:19
Brahadeesh
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
edited Dec 7 '18 at 17:19
Brahadeesh
6,35442363
edited Dec 7 '18 at 17:19
Brahadeesh
6,35442363
edited Dec 7 '18 at 17:19
Brahadeesh
6,35442363
6,35442363
asked Dec 7 '18 at 16:43
katerinekaterine
356
asked Dec 7 '18 at 16:43
katerinekaterine
356
asked Dec 7 '18 at 16:43
katerinekaterine
356
356
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
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
$begingroup$
@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
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
10.3k41641
$endgroup$
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
9201216
$endgroup$
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 |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030107%2fwhat-is-the-mathematical-word-to-describe-when-two-objects-can-be-transformed-to%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
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
10.3k41641
$endgroup$
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
10.3k41641
$endgroup$
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
10.3k41641
$endgroup$
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
10.3k41641
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
answered Dec 7 '18 at 16:53
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.3k41641
10.3k41641
add a comment |
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
9201216
$endgroup$
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
9201216
$endgroup$
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
9201216
$endgroup$
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
9201216
edited Dec 7 '18 at 17:21
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9201216
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9201216
answered Dec 7 '18 at 16:52
DreamConspiracyDreamConspiracy
9201216
9201216
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030107%2fwhat-is-the-mathematical-word-to-describe-when-two-objects-can-be-transformed-to%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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