Cardinality of dense set related to cardinality of discrete set
$begingroup$
Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?
general-topology topological-vector-spaces
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?
general-topology topological-vector-spaces
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?
general-topology topological-vector-spaces
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
$endgroup$
Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?
general-topology topological-vector-spaces
general-topology topological-vector-spaces
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
edited Jan 5 at 22:53
Stoyan Apostolov
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
asked Jan 5 at 22:42
Stoyan ApostolovStoyan Apostolov
39229
39229
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.
In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
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: "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%2f3063276%2fcardinality-of-dense-set-related-to-cardinality-of-discrete-set%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
$begingroup$
In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.
In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
add a comment |
$begingroup$
In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.
In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
add a comment |
$begingroup$
In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.
In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.
In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
answered Jan 5 at 22:50
Henno BrandsmaHenno Brandsma
116k349127
116k349127
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.
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%2f3063276%2fcardinality-of-dense-set-related-to-cardinality-of-discrete-set%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
