Product of paracompact spaces
$begingroup$
I know that the product of a compact space and a paracompact space is paracompact, and that in general the product of two paracompact spaces are not paracompact.
Question: Is there a weakest condition on a space $Y$ such that $X times Y$ is paracompact Hausdorff for every paracompact Hausdorff space $X$??
general-topology paracompactness
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
$endgroup$
add a comment |
$begingroup$
I know that the product of a compact space and a paracompact space is paracompact, and that in general the product of two paracompact spaces are not paracompact.
Question: Is there a weakest condition on a space $Y$ such that $X times Y$ is paracompact Hausdorff for every paracompact Hausdorff space $X$??
general-topology paracompactness
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
$endgroup$
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23
add a comment |
$begingroup$
I know that the product of a compact space and a paracompact space is paracompact, and that in general the product of two paracompact spaces are not paracompact.
Question: Is there a weakest condition on a space $Y$ such that $X times Y$ is paracompact Hausdorff for every paracompact Hausdorff space $X$??
general-topology paracompactness
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
$endgroup$
I know that the product of a compact space and a paracompact space is paracompact, and that in general the product of two paracompact spaces are not paracompact.
Question: Is there a weakest condition on a space $Y$ such that $X times Y$ is paracompact Hausdorff for every paracompact Hausdorff space $X$??
general-topology paracompactness
general-topology paracompactness
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
edited Dec 9 '18 at 3:19
Eric Wofsey
186k14214341
186k14214341
asked Feb 21 '15 at 0:08
user171326
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23
add a comment |
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
So you are looking for conditions $P$ such that, for all $X$ paracompact Hausdorff and all $Y$ with condition $P$, $X times Y$ is paracompact Hausdorff.
Clearly such a condition $P$ needs to imply Hausdorff paracompactness itself (but $P$ = "paracompact Hausdorff" does not work, as you stated already, and as the Sorgenfrey line shows).
This paper by Suzuki already mentions a few $P$ that work:
- $P$ = "Y paracompact Hausdorff, and $Y$ is a countable union of locally compact closed subsets" (due to K. Morita)
- $P$ = "$Y$ is Hausdorff and the closed continuous image of a locally compact Hausdorff paracompact space."
He also proves a more general $P$ than both of these. Morita already showed that for a metric space $Y$ it is in fact necessary and sufficient to be a union of a sigma-locally finite collection of compact subsets in order to have the property that the product with any
paracompact space is again paracompact, which is closely related to the Michael line example (which shows that the irrationals times a paracompact space need not be paracompact).
This paper has even more conditions and a nice survey.
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
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%2f1158251%2fproduct-of-paracompact-spaces%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
$begingroup$
So you are looking for conditions $P$ such that, for all $X$ paracompact Hausdorff and all $Y$ with condition $P$, $X times Y$ is paracompact Hausdorff.
Clearly such a condition $P$ needs to imply Hausdorff paracompactness itself (but $P$ = "paracompact Hausdorff" does not work, as you stated already, and as the Sorgenfrey line shows).
This paper by Suzuki already mentions a few $P$ that work:
- $P$ = "Y paracompact Hausdorff, and $Y$ is a countable union of locally compact closed subsets" (due to K. Morita)
- $P$ = "$Y$ is Hausdorff and the closed continuous image of a locally compact Hausdorff paracompact space."
He also proves a more general $P$ than both of these. Morita already showed that for a metric space $Y$ it is in fact necessary and sufficient to be a union of a sigma-locally finite collection of compact subsets in order to have the property that the product with any
paracompact space is again paracompact, which is closely related to the Michael line example (which shows that the irrationals times a paracompact space need not be paracompact).
This paper has even more conditions and a nice survey.
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
add a comment |
$begingroup$
So you are looking for conditions $P$ such that, for all $X$ paracompact Hausdorff and all $Y$ with condition $P$, $X times Y$ is paracompact Hausdorff.
Clearly such a condition $P$ needs to imply Hausdorff paracompactness itself (but $P$ = "paracompact Hausdorff" does not work, as you stated already, and as the Sorgenfrey line shows).
This paper by Suzuki already mentions a few $P$ that work:
- $P$ = "Y paracompact Hausdorff, and $Y$ is a countable union of locally compact closed subsets" (due to K. Morita)
- $P$ = "$Y$ is Hausdorff and the closed continuous image of a locally compact Hausdorff paracompact space."
He also proves a more general $P$ than both of these. Morita already showed that for a metric space $Y$ it is in fact necessary and sufficient to be a union of a sigma-locally finite collection of compact subsets in order to have the property that the product with any
paracompact space is again paracompact, which is closely related to the Michael line example (which shows that the irrationals times a paracompact space need not be paracompact).
This paper has even more conditions and a nice survey.
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
add a comment |
$begingroup$
So you are looking for conditions $P$ such that, for all $X$ paracompact Hausdorff and all $Y$ with condition $P$, $X times Y$ is paracompact Hausdorff.
Clearly such a condition $P$ needs to imply Hausdorff paracompactness itself (but $P$ = "paracompact Hausdorff" does not work, as you stated already, and as the Sorgenfrey line shows).
This paper by Suzuki already mentions a few $P$ that work:
- $P$ = "Y paracompact Hausdorff, and $Y$ is a countable union of locally compact closed subsets" (due to K. Morita)
- $P$ = "$Y$ is Hausdorff and the closed continuous image of a locally compact Hausdorff paracompact space."
He also proves a more general $P$ than both of these. Morita already showed that for a metric space $Y$ it is in fact necessary and sufficient to be a union of a sigma-locally finite collection of compact subsets in order to have the property that the product with any
paracompact space is again paracompact, which is closely related to the Michael line example (which shows that the irrationals times a paracompact space need not be paracompact).
This paper has even more conditions and a nice survey.
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
$endgroup$
So you are looking for conditions $P$ such that, for all $X$ paracompact Hausdorff and all $Y$ with condition $P$, $X times Y$ is paracompact Hausdorff.
Clearly such a condition $P$ needs to imply Hausdorff paracompactness itself (but $P$ = "paracompact Hausdorff" does not work, as you stated already, and as the Sorgenfrey line shows).
This paper by Suzuki already mentions a few $P$ that work:
- $P$ = "Y paracompact Hausdorff, and $Y$ is a countable union of locally compact closed subsets" (due to K. Morita)
- $P$ = "$Y$ is Hausdorff and the closed continuous image of a locally compact Hausdorff paracompact space."
He also proves a more general $P$ than both of these. Morita already showed that for a metric space $Y$ it is in fact necessary and sufficient to be a union of a sigma-locally finite collection of compact subsets in order to have the property that the product with any
paracompact space is again paracompact, which is closely related to the Michael line example (which shows that the irrationals times a paracompact space need not be paracompact).
This paper has even more conditions and a nice survey.
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
edited Feb 21 '15 at 12:32
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
answered Feb 21 '15 at 12:20
Henno BrandsmaHenno Brandsma
109k347115
109k347115
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
add a comment |
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
$begingroup$
@N.H. Glad to help. I believe there has been some more developments (using topological games) as well. I'm not sure there is an exact characterisation yet..
$endgroup$
– Henno Brandsma
Feb 21 '15 at 17:47
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%2f1158251%2fproduct-of-paracompact-spaces%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
$begingroup$
Do you mean that $X$ is given and we can choose the condition on $Y$ dependent on $X$? Or, do you mean that: what is a condition $P$ such that for all $X$ paracompact (Hausdorff?) and all $Y$ with condition $P$, we have that $X times Y$ is paracompact?
$endgroup$
– Henno Brandsma
Feb 21 '15 at 11:20
$begingroup$
I mean the second formulation : is there a condition $P$ such that ... (Yes for Hausdorff !!)
$endgroup$
– user171326
Feb 21 '15 at 11:23