Can a noncompact metric space have a maximal metrizable Hausdorff compactification?
We know the Stone-Čech compactification $(h, beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,gamma X$) is any other Hausdorff compactification, then there is a continuous map $f:beta X
rightarrow gamma X$ such that $fh=k$.
What about if I restrict to considering just all metrizable Hausdorff compactifications? To fix ideas, say $X$ is a noncompact metric space. (In particular $beta X$ will not be metrizable.)
Is there a maximal metrizable Hausdorff compactification of $X$?
I am guessing no: if we're given any metrizable Hausdorff compactification of noncompact metric space $X$, we can make another metrizable Hausdorff compactification that is larger (in the sense of the first paragraph above).
general-topology metric-spaces compactness compactification
add a comment |
We know the Stone-Čech compactification $(h, beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,gamma X$) is any other Hausdorff compactification, then there is a continuous map $f:beta X
rightarrow gamma X$ such that $fh=k$.
What about if I restrict to considering just all metrizable Hausdorff compactifications? To fix ideas, say $X$ is a noncompact metric space. (In particular $beta X$ will not be metrizable.)
Is there a maximal metrizable Hausdorff compactification of $X$?
I am guessing no: if we're given any metrizable Hausdorff compactification of noncompact metric space $X$, we can make another metrizable Hausdorff compactification that is larger (in the sense of the first paragraph above).
general-topology metric-spaces compactness compactification
add a comment |
We know the Stone-Čech compactification $(h, beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,gamma X$) is any other Hausdorff compactification, then there is a continuous map $f:beta X
rightarrow gamma X$ such that $fh=k$.
What about if I restrict to considering just all metrizable Hausdorff compactifications? To fix ideas, say $X$ is a noncompact metric space. (In particular $beta X$ will not be metrizable.)
Is there a maximal metrizable Hausdorff compactification of $X$?
I am guessing no: if we're given any metrizable Hausdorff compactification of noncompact metric space $X$, we can make another metrizable Hausdorff compactification that is larger (in the sense of the first paragraph above).
general-topology metric-spaces compactness compactification
We know the Stone-Čech compactification $(h, beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,gamma X$) is any other Hausdorff compactification, then there is a continuous map $f:beta X
rightarrow gamma X$ such that $fh=k$.
What about if I restrict to considering just all metrizable Hausdorff compactifications? To fix ideas, say $X$ is a noncompact metric space. (In particular $beta X$ will not be metrizable.)
Is there a maximal metrizable Hausdorff compactification of $X$?
I am guessing no: if we're given any metrizable Hausdorff compactification of noncompact metric space $X$, we can make another metrizable Hausdorff compactification that is larger (in the sense of the first paragraph above).
general-topology metric-spaces compactness compactification
general-topology metric-spaces compactness compactification
edited Nov 28 '18 at 21:06
Eric Wofsey
180k12207335
180k12207335
asked Nov 28 '18 at 20:02
SSF
362110
362110
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
No. Let $X$ be a noncompact metric space and let $Y$ be a metrizable compactification of $X$. Pick a point $yin Y$ and a sequence $(x_n)$ of distinct points in $X$ converging to $y$. Let $A$ consist of the points $x_n$ for all even $n$ and let $B$ consist of the points $x_n$ for all odd $n$. Then $A$ and $B$ are closed and disjoint in $X$, so there exists a continuous $f:Xto[0,1]$ such that $f$ maps $A$ to $0$ and $B$ to $1$. This map $f$ does not extend continuously to $Y$.
We can now embed $X$ in $Ytimes [0,1]$ by using the given embedding $Xto Y$ on the first coordinate and $f$ on the second coordinate. The closure of the image of $X$ in $Ytimes[0,1]$ is then a metrizable compactification of $X$, which is strictly larger than $Y$ since $f$ extends continuously to it.
For another perspective on this, note that the algebra $C_b(X)$ of bounded continuous functions on $X$ can canonically be identified with $C(beta X)$, and an arbitrary compactification of $X$ just corresponds to a closed subalgebra of $C_b(X)$ which separates points from closed sets on $X$. Moreover, a compactification is metrizable iff the corresponding subalgebra is separable. "Larger" compactifications in your sense just corresponds to the inclusion order on subalgebras.
So, given any metrizable compactification corresponding to a separable subalgebra $C(Y)subset C_b(X)$, you can get a larger separable subalgebra by just taking some $fin C_b(X)setminus C(Y)$ and taking the closed subalgebra generated by $f$ and $C(Y)$.
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
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%2f3017623%2fcan-a-noncompact-metric-space-have-a-maximal-metrizable-hausdorff-compactificati%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
No. Let $X$ be a noncompact metric space and let $Y$ be a metrizable compactification of $X$. Pick a point $yin Y$ and a sequence $(x_n)$ of distinct points in $X$ converging to $y$. Let $A$ consist of the points $x_n$ for all even $n$ and let $B$ consist of the points $x_n$ for all odd $n$. Then $A$ and $B$ are closed and disjoint in $X$, so there exists a continuous $f:Xto[0,1]$ such that $f$ maps $A$ to $0$ and $B$ to $1$. This map $f$ does not extend continuously to $Y$.
We can now embed $X$ in $Ytimes [0,1]$ by using the given embedding $Xto Y$ on the first coordinate and $f$ on the second coordinate. The closure of the image of $X$ in $Ytimes[0,1]$ is then a metrizable compactification of $X$, which is strictly larger than $Y$ since $f$ extends continuously to it.
For another perspective on this, note that the algebra $C_b(X)$ of bounded continuous functions on $X$ can canonically be identified with $C(beta X)$, and an arbitrary compactification of $X$ just corresponds to a closed subalgebra of $C_b(X)$ which separates points from closed sets on $X$. Moreover, a compactification is metrizable iff the corresponding subalgebra is separable. "Larger" compactifications in your sense just corresponds to the inclusion order on subalgebras.
So, given any metrizable compactification corresponding to a separable subalgebra $C(Y)subset C_b(X)$, you can get a larger separable subalgebra by just taking some $fin C_b(X)setminus C(Y)$ and taking the closed subalgebra generated by $f$ and $C(Y)$.
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
add a comment |
No. Let $X$ be a noncompact metric space and let $Y$ be a metrizable compactification of $X$. Pick a point $yin Y$ and a sequence $(x_n)$ of distinct points in $X$ converging to $y$. Let $A$ consist of the points $x_n$ for all even $n$ and let $B$ consist of the points $x_n$ for all odd $n$. Then $A$ and $B$ are closed and disjoint in $X$, so there exists a continuous $f:Xto[0,1]$ such that $f$ maps $A$ to $0$ and $B$ to $1$. This map $f$ does not extend continuously to $Y$.
We can now embed $X$ in $Ytimes [0,1]$ by using the given embedding $Xto Y$ on the first coordinate and $f$ on the second coordinate. The closure of the image of $X$ in $Ytimes[0,1]$ is then a metrizable compactification of $X$, which is strictly larger than $Y$ since $f$ extends continuously to it.
For another perspective on this, note that the algebra $C_b(X)$ of bounded continuous functions on $X$ can canonically be identified with $C(beta X)$, and an arbitrary compactification of $X$ just corresponds to a closed subalgebra of $C_b(X)$ which separates points from closed sets on $X$. Moreover, a compactification is metrizable iff the corresponding subalgebra is separable. "Larger" compactifications in your sense just corresponds to the inclusion order on subalgebras.
So, given any metrizable compactification corresponding to a separable subalgebra $C(Y)subset C_b(X)$, you can get a larger separable subalgebra by just taking some $fin C_b(X)setminus C(Y)$ and taking the closed subalgebra generated by $f$ and $C(Y)$.
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
add a comment |
No. Let $X$ be a noncompact metric space and let $Y$ be a metrizable compactification of $X$. Pick a point $yin Y$ and a sequence $(x_n)$ of distinct points in $X$ converging to $y$. Let $A$ consist of the points $x_n$ for all even $n$ and let $B$ consist of the points $x_n$ for all odd $n$. Then $A$ and $B$ are closed and disjoint in $X$, so there exists a continuous $f:Xto[0,1]$ such that $f$ maps $A$ to $0$ and $B$ to $1$. This map $f$ does not extend continuously to $Y$.
We can now embed $X$ in $Ytimes [0,1]$ by using the given embedding $Xto Y$ on the first coordinate and $f$ on the second coordinate. The closure of the image of $X$ in $Ytimes[0,1]$ is then a metrizable compactification of $X$, which is strictly larger than $Y$ since $f$ extends continuously to it.
For another perspective on this, note that the algebra $C_b(X)$ of bounded continuous functions on $X$ can canonically be identified with $C(beta X)$, and an arbitrary compactification of $X$ just corresponds to a closed subalgebra of $C_b(X)$ which separates points from closed sets on $X$. Moreover, a compactification is metrizable iff the corresponding subalgebra is separable. "Larger" compactifications in your sense just corresponds to the inclusion order on subalgebras.
So, given any metrizable compactification corresponding to a separable subalgebra $C(Y)subset C_b(X)$, you can get a larger separable subalgebra by just taking some $fin C_b(X)setminus C(Y)$ and taking the closed subalgebra generated by $f$ and $C(Y)$.
No. Let $X$ be a noncompact metric space and let $Y$ be a metrizable compactification of $X$. Pick a point $yin Y$ and a sequence $(x_n)$ of distinct points in $X$ converging to $y$. Let $A$ consist of the points $x_n$ for all even $n$ and let $B$ consist of the points $x_n$ for all odd $n$. Then $A$ and $B$ are closed and disjoint in $X$, so there exists a continuous $f:Xto[0,1]$ such that $f$ maps $A$ to $0$ and $B$ to $1$. This map $f$ does not extend continuously to $Y$.
We can now embed $X$ in $Ytimes [0,1]$ by using the given embedding $Xto Y$ on the first coordinate and $f$ on the second coordinate. The closure of the image of $X$ in $Ytimes[0,1]$ is then a metrizable compactification of $X$, which is strictly larger than $Y$ since $f$ extends continuously to it.
For another perspective on this, note that the algebra $C_b(X)$ of bounded continuous functions on $X$ can canonically be identified with $C(beta X)$, and an arbitrary compactification of $X$ just corresponds to a closed subalgebra of $C_b(X)$ which separates points from closed sets on $X$. Moreover, a compactification is metrizable iff the corresponding subalgebra is separable. "Larger" compactifications in your sense just corresponds to the inclusion order on subalgebras.
So, given any metrizable compactification corresponding to a separable subalgebra $C(Y)subset C_b(X)$, you can get a larger separable subalgebra by just taking some $fin C_b(X)setminus C(Y)$ and taking the closed subalgebra generated by $f$ and $C(Y)$.
edited Nov 28 '18 at 21:06
answered Nov 28 '18 at 21:01
Eric Wofsey
180k12207335
180k12207335
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
add a comment |
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
May I ask why the closure of the image of $X$ in $Y×[0,1]$ is a compactification of $X$? I've gone through the other details, but in particular why is the inverse function of the embedding continuous?
– SSF
Dec 8 '18 at 2:20
1
1
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
The inverse function is the composition of the projection $Ytimes [0,1]to Y$ and the inverse of the embedding $Xto Y$.
– Eric Wofsey
Dec 8 '18 at 2:24
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%2f3017623%2fcan-a-noncompact-metric-space-have-a-maximal-metrizable-hausdorff-compactificati%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