Mixing
Is $[0,1]^A$ not sequentially compact for any uncountable $A$?
$begingroup$
We know ${0,1}^mathbb{N}$ is sequentially compact, but ${0,1}^mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a continuum. So not assuming the continuum hypothesis, is it still true that ${0,1}^A$ is not sequentially compact for any uncountable $A$?
compactness product-space
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
$endgroup$
add a comment |
$begingroup$
We know ${0,1}^mathbb{N}$ is sequentially compact, but ${0,1}^mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a continuum. So not assuming the continuum hypothesis, is it still true that ${0,1}^A$ is not sequentially compact for any uncountable $A$?
compactness product-space
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
$endgroup$
2
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08
add a comment |
$begingroup$
We know ${0,1}^mathbb{N}$ is sequentially compact, but ${0,1}^mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a continuum. So not assuming the continuum hypothesis, is it still true that ${0,1}^A$ is not sequentially compact for any uncountable $A$?
compactness product-space
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
$endgroup$
We know ${0,1}^mathbb{N}$ is sequentially compact, but ${0,1}^mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a continuum. So not assuming the continuum hypothesis, is it still true that ${0,1}^A$ is not sequentially compact for any uncountable $A$?
compactness product-space
compactness product-space
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
asked Jan 20 at 5:33
SmileyCraftSmileyCraft
3,476518
3,476518
2
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08
add a comment |
2
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08
2
2
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08
add a comment |
0
active
oldest
votes
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%2f3080228%2fis-0-1a-not-sequentially-compact-for-any-uncountable-a%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3080228%2fis-0-1a-not-sequentially-compact-for-any-uncountable-a%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
$begingroup$
Perhaps this is on point.
$endgroup$
– David Mitra
Jan 20 at 7:08