Prove that we can write any subset of Real numbers as a Union of closed subsets, or intersect of open...
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
$endgroup$
add a comment |
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
$endgroup$
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
$endgroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
edited Dec 18 '18 at 15:38
Arman_jr
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
285
285
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
$endgroup$
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%2f3045259%2fprove-that-we-can-write-any-subset-of-real-numbers-as-a-union-of-closed-subsets%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
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
$endgroup$
add a comment |
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
$endgroup$
add a comment |
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
$endgroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
952114
952114
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%2f3045259%2fprove-that-we-can-write-any-subset-of-real-numbers-as-a-union-of-closed-subsets%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 union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09