Is it possible that $xin overlineAcap B$ for $xin A$ $B$ - open The Next CEO of Stack OverflowWhat conditions are sufficient for “Basically disconnectedness implies Extremally disconnectedness”?Is every Extremally Disconnected Hausdorff Space Regular?A topological space is extremally disconnected iff every two disjoint open sets have disjoint closuresA is a open set $ iff (forall X) (A cap overlineX subset overlineA cap X )$Let $Usubseteq X$ be open and $Ssubseteq X$ a set such that $Scap U$ is closed in $U$. Does $barScap U=Scap U$?If $overline Acap B = Acap overline B = varnothing$, $Acup B$ is disconnected.Quotients of extremally disconnected compact Hausdorff spacesExtremally disconnected space is normal$X$ is basically disconnected if every cozero-set has an open closure.Prove that If B is open, then $overlineA cap B subset overlineA cap B$
Can Sri Krishna be called 'a person'?
How to show a landlord what we have in savings?
Calculating discount not working
Why did early computer designers eschew integers?
"Eavesdropping" vs "Listen in on"
Man transported from Alternate World into ours by a Neutrino Detector
MT "will strike" & LXX "will watch carefully" (Gen 3:15)?
How to unfasten electrical subpanel attached with ramset
Could you use a laser beam as a modulated carrier wave for radio signal?
Planeswalker Ability and Death Timing
Strange use of "whether ... than ..." in official text
Masking layers by a vector polygon layer in QGIS
How should I connect my cat5 cable to connectors having an orange-green line?
Small nick on power cord from an electric alarm clock, and copper wiring exposed but intact
Is it okay to majorly distort historical facts while writing a fiction story?
logical reads on global temp table, but not on session-level temp table
Why can't we say "I have been having a dog"?
Compensation for working overtime on Saturdays
Raspberry pi 3 B with Ubuntu 18.04 server arm64: what pi version
How can I separate the number from the unit in argument?
How seriously should I take size and weight limits of hand luggage?
The sum of any ten consecutive numbers from a fibonacci sequence is divisible by 11
What day is it again?
Why does freezing point matter when picking cooler ice packs?
Is it possible that $xin overlineAcap B$ for $xin A$ $B$ - open
The Next CEO of Stack OverflowWhat conditions are sufficient for “Basically disconnectedness implies Extremally disconnectedness”?Is every Extremally Disconnected Hausdorff Space Regular?A topological space is extremally disconnected iff every two disjoint open sets have disjoint closuresA is a open set $ iff (forall X) (A cap overlineX subset overlineA cap X )$Let $Usubseteq X$ be open and $Ssubseteq X$ a set such that $Scap U$ is closed in $U$. Does $barScap U=Scap U$?If $overline Acap B = Acap overline B = varnothing$, $Acup B$ is disconnected.Quotients of extremally disconnected compact Hausdorff spacesExtremally disconnected space is normal$X$ is basically disconnected if every cozero-set has an open closure.Prove that If B is open, then $overlineA cap B subset overlineA cap B$
$begingroup$
Is the following possible:
For a Hausdorff space (don't know if that's needed), let $xin A$ $B$ for $A,Bsubseteq_opX$. Is it possible that $xinoverlineAcap B$?
What about if $X$ is extremally disconnected (that is, the closure of each open set is open)
general-topology
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
$endgroup$
add a comment |
$begingroup$
Is the following possible:
For a Hausdorff space (don't know if that's needed), let $xin A$ $B$ for $A,Bsubseteq_opX$. Is it possible that $xinoverlineAcap B$?
What about if $X$ is extremally disconnected (that is, the closure of each open set is open)
general-topology
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
$endgroup$
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44
add a comment |
$begingroup$
Is the following possible:
For a Hausdorff space (don't know if that's needed), let $xin A$ $B$ for $A,Bsubseteq_opX$. Is it possible that $xinoverlineAcap B$?
What about if $X$ is extremally disconnected (that is, the closure of each open set is open)
general-topology
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
$endgroup$
Is the following possible:
For a Hausdorff space (don't know if that's needed), let $xin A$ $B$ for $A,Bsubseteq_opX$. Is it possible that $xinoverlineAcap B$?
What about if $X$ is extremally disconnected (that is, the closure of each open set is open)
general-topology
general-topology
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
edited Mar 20 at 8:35
YuiTo Cheng
2,1863937
2,1863937
asked Mar 18 at 18:23
user3701033user3701033
1519
asked Mar 18 at 18:23
user3701033user3701033
1519
asked Mar 18 at 18:23
user3701033user3701033
1519
1519
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44
add a comment |
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes this is possible. In general, whenever $(A cap B)$ is dense in some neighbourhood of a point in $A setminus B$, $cl(A cap B)$ wll contain that point.
Suppose $A = X = [0,1]$. Let $q_n$ be an enumeration of the rationals in $[0,1]$. For some $epsilon < 1$, let $B_2^-nepsilon(q_n)$ be the open ball of diameter $2^-nepsilon$ around $q_n$, and let $B = bigcup_nB_2^-nepsilon(q_n)$.
$B$ is open and dense in $A$, so $cl(B) = A$. And the Lebesgue measure of $B$, $mu(B) leq sum_n geq 12^-nepsilon = epsilon < 1 = mu(A)$, so $B$ is a proper subset of $A$, i.e., $A setminus B$ is non-empty.
This shows that $A cap B = B$, but $B$ is dense in $A$ so $cl(A cap B) = A$.
answered Mar 18 at 20:55
vxnturevxnture
39910
$endgroup$
add a comment |
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%2f3153124%2fis-it-possible-that-x-in-overlinea-cap-b-for-x-in-a-b-open%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
$begingroup$
Yes this is possible. In general, whenever $(A cap B)$ is dense in some neighbourhood of a point in $A setminus B$, $cl(A cap B)$ wll contain that point.
Suppose $A = X = [0,1]$. Let $q_n$ be an enumeration of the rationals in $[0,1]$. For some $epsilon < 1$, let $B_2^-nepsilon(q_n)$ be the open ball of diameter $2^-nepsilon$ around $q_n$, and let $B = bigcup_nB_2^-nepsilon(q_n)$.
$B$ is open and dense in $A$, so $cl(B) = A$. And the Lebesgue measure of $B$, $mu(B) leq sum_n geq 12^-nepsilon = epsilon < 1 = mu(A)$, so $B$ is a proper subset of $A$, i.e., $A setminus B$ is non-empty.
This shows that $A cap B = B$, but $B$ is dense in $A$ so $cl(A cap B) = A$.
answered Mar 18 at 20:55
vxnturevxnture
39910
$endgroup$
add a comment |
$begingroup$
Yes this is possible. In general, whenever $(A cap B)$ is dense in some neighbourhood of a point in $A setminus B$, $cl(A cap B)$ wll contain that point.
Suppose $A = X = [0,1]$. Let $q_n$ be an enumeration of the rationals in $[0,1]$. For some $epsilon < 1$, let $B_2^-nepsilon(q_n)$ be the open ball of diameter $2^-nepsilon$ around $q_n$, and let $B = bigcup_nB_2^-nepsilon(q_n)$.
$B$ is open and dense in $A$, so $cl(B) = A$. And the Lebesgue measure of $B$, $mu(B) leq sum_n geq 12^-nepsilon = epsilon < 1 = mu(A)$, so $B$ is a proper subset of $A$, i.e., $A setminus B$ is non-empty.
This shows that $A cap B = B$, but $B$ is dense in $A$ so $cl(A cap B) = A$.
answered Mar 18 at 20:55
vxnturevxnture
39910
$endgroup$
add a comment |
$begingroup$
Yes this is possible. In general, whenever $(A cap B)$ is dense in some neighbourhood of a point in $A setminus B$, $cl(A cap B)$ wll contain that point.
Suppose $A = X = [0,1]$. Let $q_n$ be an enumeration of the rationals in $[0,1]$. For some $epsilon < 1$, let $B_2^-nepsilon(q_n)$ be the open ball of diameter $2^-nepsilon$ around $q_n$, and let $B = bigcup_nB_2^-nepsilon(q_n)$.
$B$ is open and dense in $A$, so $cl(B) = A$. And the Lebesgue measure of $B$, $mu(B) leq sum_n geq 12^-nepsilon = epsilon < 1 = mu(A)$, so $B$ is a proper subset of $A$, i.e., $A setminus B$ is non-empty.
This shows that $A cap B = B$, but $B$ is dense in $A$ so $cl(A cap B) = A$.
answered Mar 18 at 20:55
vxnturevxnture
39910
$endgroup$
Yes this is possible. In general, whenever $(A cap B)$ is dense in some neighbourhood of a point in $A setminus B$, $cl(A cap B)$ wll contain that point.
Suppose $A = X = [0,1]$. Let $q_n$ be an enumeration of the rationals in $[0,1]$. For some $epsilon < 1$, let $B_2^-nepsilon(q_n)$ be the open ball of diameter $2^-nepsilon$ around $q_n$, and let $B = bigcup_nB_2^-nepsilon(q_n)$.
$B$ is open and dense in $A$, so $cl(B) = A$. And the Lebesgue measure of $B$, $mu(B) leq sum_n geq 12^-nepsilon = epsilon < 1 = mu(A)$, so $B$ is a proper subset of $A$, i.e., $A setminus B$ is non-empty.
This shows that $A cap B = B$, but $B$ is dense in $A$ so $cl(A cap B) = A$.
answered Mar 18 at 20:55
vxnturevxnture
39910
answered Mar 18 at 20:55
vxnturevxnture
39910
answered Mar 18 at 20:55
vxnturevxnture
39910
answered Mar 18 at 20:55
vxnturevxnture
39910
39910
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%2f3153124%2fis-it-possible-that-x-in-overlinea-cap-b-for-x-in-a-b-open%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
$begingroup$
What have you tried so far? Hint: When (if at all) is it possible that $cl(A cap B)$ contains a point outside $B$?
$endgroup$
– vxnture
Mar 18 at 19:35
$begingroup$
It is equivalent to proving that cl$(Acap B) subseteq$ cl$(A)cap B$
$endgroup$
– user3701033
Mar 18 at 20:44