Fixed point property and interval topology
$begingroup$
Given a poset $(P,leq)$ the interval topology $tau_i(P)$ on $P$ is generated by
$${Psetminusdownarrow x : xin P} cup {Psetminusuparrow x : xin P},$$
where $downarrow x = {yin P: yleq x}$ and $uparrow x = {yin P: ygeq x}$.
We say that a poset $(P,leq)$ has the fixed point property (FPP) if for every order-preserving map $f:Pto P$ there is $xin P$ such that $f(x) = x$.
Question. If $(P,leq)$ has the (FPP), does every continuous map from the topological space $(P,tau_i(P))$ to itself have a fixed point?
gn.general-topology order-theory
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
$endgroup$
add a comment |
$begingroup$
Given a poset $(P,leq)$ the interval topology $tau_i(P)$ on $P$ is generated by
$${Psetminusdownarrow x : xin P} cup {Psetminusuparrow x : xin P},$$
where $downarrow x = {yin P: yleq x}$ and $uparrow x = {yin P: ygeq x}$.
We say that a poset $(P,leq)$ has the fixed point property (FPP) if for every order-preserving map $f:Pto P$ there is $xin P$ such that $f(x) = x$.
Question. If $(P,leq)$ has the (FPP), does every continuous map from the topological space $(P,tau_i(P))$ to itself have a fixed point?
gn.general-topology order-theory
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
$endgroup$
1
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30
add a comment |
$begingroup$
Given a poset $(P,leq)$ the interval topology $tau_i(P)$ on $P$ is generated by
$${Psetminusdownarrow x : xin P} cup {Psetminusuparrow x : xin P},$$
where $downarrow x = {yin P: yleq x}$ and $uparrow x = {yin P: ygeq x}$.
We say that a poset $(P,leq)$ has the fixed point property (FPP) if for every order-preserving map $f:Pto P$ there is $xin P$ such that $f(x) = x$.
Question. If $(P,leq)$ has the (FPP), does every continuous map from the topological space $(P,tau_i(P))$ to itself have a fixed point?
gn.general-topology order-theory
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
$endgroup$
Given a poset $(P,leq)$ the interval topology $tau_i(P)$ on $P$ is generated by
$${Psetminusdownarrow x : xin P} cup {Psetminusuparrow x : xin P},$$
where $downarrow x = {yin P: yleq x}$ and $uparrow x = {yin P: ygeq x}$.
We say that a poset $(P,leq)$ has the fixed point property (FPP) if for every order-preserving map $f:Pto P$ there is $xin P$ such that $f(x) = x$.
Question. If $(P,leq)$ has the (FPP), does every continuous map from the topological space $(P,tau_i(P))$ to itself have a fixed point?
gn.general-topology order-theory
gn.general-topology order-theory
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
asked Feb 1 at 9:12
Dominic van der ZypenDominic van der Zypen
15k43280
15k43280
1
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30
add a comment |
1
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30
1
1
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
$endgroup$
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: "504"
};
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%2fmathoverflow.net%2fquestions%2f322202%2ffixed-point-property-and-interval-topology%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
$begingroup$
Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
$endgroup$
add a comment |
$begingroup$
Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
$endgroup$
add a comment |
$begingroup$
Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
$endgroup$
Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
edited Feb 1 at 16:09
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
answered Feb 1 at 10:33
WojowuWojowu
6,65912851
6,65912851
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f322202%2ffixed-point-property-and-interval-topology%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
1
$begingroup$
How about total order on two elements?
$endgroup$
– Wojowu
Feb 1 at 9:35
$begingroup$
Right :) you can post this as an answer, and I'll accept it - or I delete the question. Your choice!
$endgroup$
– Dominic van der Zypen
Feb 1 at 10:30