Are $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| x^4 , |y| < 10 right} $ homeomorphic?
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
add a comment |
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
edited Nov 19 at 9:15
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
1608
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
1 Answer
1
active
oldest
votes
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
11.3k1629
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%2f3003604%2fare-left-x-y-in-mathbbr2-y-x2-y-10-right-left%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
11.3k1629
add a comment |
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
11.3k1629
add a comment |
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
11.3k1629
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
11.3k1629
answered Nov 19 at 9:29
freakish
11.3k1629
answered Nov 19 at 9:29
freakish
11.3k1629
answered Nov 19 at 9:29
freakish
11.3k1629
11.3k1629
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3003604%2fare-left-x-y-in-mathbbr2-y-x2-y-10-right-left%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
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19