What are some nice characterizations of $mathfrak{c}$-compact spaces?
$begingroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
$endgroup$
add a comment |
$begingroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
$endgroup$
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
add a comment |
$begingroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
$endgroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,64622467
8,64622467
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
add a comment |
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
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%2f3044933%2fwhat-are-some-nice-characterizations-of-mathfrakc-compact-spaces%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%2f3044933%2fwhat-are-some-nice-characterizations-of-mathfrakc-compact-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
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18