Is a weak basis of a topological space a basis and vice versa? (Kechris' book)Definition of a basis for a particular topological spaceBases and open covers of a topological spaceExercise of comager set and the space $C([0,1])$Question about of Baire property and Baire spaceFinding a neighborhhod basis of a specific topological space.Name for topological space where every neighborhood of some point is open?General Topology and Basis definitionTopological basis vs local basis.Basis of topological spaceBasis of topological space and topological subspace
Maximum likelihood parameters deviate from posterior distributions
Watching something be written to a file live with tail
What does it mean to describe someone as a butt steak?
Malformed Address '10.10.21.08/24', must be X.X.X.X/NN or
Is it possible to do 50 km distance without any previous training?
RSA: Danger of using p to create q
How can I prevent hyper evolved versions of regular creatures from wiping out their cousins?
I'm flying to France today and my passport expires in less than 2 months
Why are electrically insulating heatsinks so rare? Is it just cost?
When a company launches a new product do they "come out" with a new product or do they "come up" with a new product?
Paid for article while in US on F-1 visa?
What's that red-plus icon near a text?
How does quantile regression compare to logistic regression with the variable split at the quantile?
What are these boxed doors outside store fronts in New York?
How much of data wrangling is a data scientist's job?
"You are your self first supporter", a more proper way to say it
What does "Puller Prush Person" mean?
What is a clear way to write a bar that has an extra beat?
Did Shadowfax go to Valinor?
Can an x86 CPU running in real mode be considered to be basically an 8086 CPU?
How old can references or sources in a thesis be?
Can a monk's single staff be considered dual wielded, as per the Dual Wielder feat?
How much RAM could one put in a typical 80386 setup?
Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)
Is a weak basis of a topological space a basis and vice versa? (Kechris' book)
Definition of a basis for a particular topological spaceBases and open covers of a topological spaceExercise of comager set and the space $C([0,1])$Question about of Baire property and Baire spaceFinding a neighborhhod basis of a specific topological space.Name for topological space where every neighborhood of some point is open?General Topology and Basis definitionTopological basis vs local basis.Basis of topological spaceBasis of topological space and topological subspace
$begingroup$
I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$):
A weak basis of a topological space $X$ is a collection of nonempty open sets s.t. every nonempty open set contains one of them.
My question is: is this definition equivalent to that of a basis for a topology?
The fact that the author gives a specific name to such a family suggests that it is not, but for every $xin X$ and for every open nhbd $U(x)$ there exists $V(x)$ in the weak basis contained in $U$. This means that a weak basis is also a covering and hence satisfies the conditions for being a basis.
Any comment is appreciated.
Thank you in advance for your help.
general-topology descriptive-set-theory
asked Mar 29 at 10:49
LBJFSLBJFS
360112
$endgroup$
add a comment |
$begingroup$
I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$):
A weak basis of a topological space $X$ is a collection of nonempty open sets s.t. every nonempty open set contains one of them.
My question is: is this definition equivalent to that of a basis for a topology?
The fact that the author gives a specific name to such a family suggests that it is not, but for every $xin X$ and for every open nhbd $U(x)$ there exists $V(x)$ in the weak basis contained in $U$. This means that a weak basis is also a covering and hence satisfies the conditions for being a basis.
Any comment is appreciated.
Thank you in advance for your help.
general-topology descriptive-set-theory
asked Mar 29 at 10:49
LBJFSLBJFS
360112
$endgroup$
1
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37
add a comment |
$begingroup$
I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$):
A weak basis of a topological space $X$ is a collection of nonempty open sets s.t. every nonempty open set contains one of them.
My question is: is this definition equivalent to that of a basis for a topology?
The fact that the author gives a specific name to such a family suggests that it is not, but for every $xin X$ and for every open nhbd $U(x)$ there exists $V(x)$ in the weak basis contained in $U$. This means that a weak basis is also a covering and hence satisfies the conditions for being a basis.
Any comment is appreciated.
Thank you in advance for your help.
general-topology descriptive-set-theory
asked Mar 29 at 10:49
LBJFSLBJFS
360112
$endgroup$
I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$):
A weak basis of a topological space $X$ is a collection of nonempty open sets s.t. every nonempty open set contains one of them.
My question is: is this definition equivalent to that of a basis for a topology?
The fact that the author gives a specific name to such a family suggests that it is not, but for every $xin X$ and for every open nhbd $U(x)$ there exists $V(x)$ in the weak basis contained in $U$. This means that a weak basis is also a covering and hence satisfies the conditions for being a basis.
Any comment is appreciated.
Thank you in advance for your help.
general-topology descriptive-set-theory
general-topology descriptive-set-theory
asked Mar 29 at 10:49
LBJFSLBJFS
360112
asked Mar 29 at 10:49
LBJFSLBJFS
360112
edited Mar 29 at 10:51
LBJFS
asked Mar 29 at 10:49
LBJFSLBJFS
360112
asked Mar 29 at 10:49
LBJFSLBJFS
360112
asked Mar 29 at 10:49
LBJFSLBJFS
360112
360112
1
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37
add a comment |
1
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37
1
1
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
While it is true that for every $x,$ any neighborhood $U(x)$ contains an element of the weak basis, say $V(x),$ we don't know that $V(x)$ is a neighborhood of $x$! All we know is that it is a subset of $U(x)$ and that it is open and nonempty. Thus, a weak basis need not cover the space, so need not be a basis.
For example, consider the topology of the empty set together with the cofinite sets (sets whose complement is finite) on the set of non-negative integers. A weak basis would be the set of cofinite sets of positive integers, but this cannot be a basis, having no neighborhood of $0.$
In general--among $T_1$ spaces, anyway--I suspect that if a space has the property that every weak basis is a basis, then the space is discrete. (The converse trivially holds.)
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
$endgroup$
add a comment |
$begingroup$
A base covers the space.
A weak base may not cover the space.
The set of all not empty, open subsets of R that exclude 0 is a weak base.
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
$endgroup$
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%2f3166999%2fis-a-weak-basis-of-a-topological-space-a-basis-and-vice-versa-kechris-book%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
While it is true that for every $x,$ any neighborhood $U(x)$ contains an element of the weak basis, say $V(x),$ we don't know that $V(x)$ is a neighborhood of $x$! All we know is that it is a subset of $U(x)$ and that it is open and nonempty. Thus, a weak basis need not cover the space, so need not be a basis.
For example, consider the topology of the empty set together with the cofinite sets (sets whose complement is finite) on the set of non-negative integers. A weak basis would be the set of cofinite sets of positive integers, but this cannot be a basis, having no neighborhood of $0.$
In general--among $T_1$ spaces, anyway--I suspect that if a space has the property that every weak basis is a basis, then the space is discrete. (The converse trivially holds.)
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
$endgroup$
add a comment |
$begingroup$
While it is true that for every $x,$ any neighborhood $U(x)$ contains an element of the weak basis, say $V(x),$ we don't know that $V(x)$ is a neighborhood of $x$! All we know is that it is a subset of $U(x)$ and that it is open and nonempty. Thus, a weak basis need not cover the space, so need not be a basis.
For example, consider the topology of the empty set together with the cofinite sets (sets whose complement is finite) on the set of non-negative integers. A weak basis would be the set of cofinite sets of positive integers, but this cannot be a basis, having no neighborhood of $0.$
In general--among $T_1$ spaces, anyway--I suspect that if a space has the property that every weak basis is a basis, then the space is discrete. (The converse trivially holds.)
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
$endgroup$
add a comment |
$begingroup$
While it is true that for every $x,$ any neighborhood $U(x)$ contains an element of the weak basis, say $V(x),$ we don't know that $V(x)$ is a neighborhood of $x$! All we know is that it is a subset of $U(x)$ and that it is open and nonempty. Thus, a weak basis need not cover the space, so need not be a basis.
For example, consider the topology of the empty set together with the cofinite sets (sets whose complement is finite) on the set of non-negative integers. A weak basis would be the set of cofinite sets of positive integers, but this cannot be a basis, having no neighborhood of $0.$
In general--among $T_1$ spaces, anyway--I suspect that if a space has the property that every weak basis is a basis, then the space is discrete. (The converse trivially holds.)
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
$endgroup$
While it is true that for every $x,$ any neighborhood $U(x)$ contains an element of the weak basis, say $V(x),$ we don't know that $V(x)$ is a neighborhood of $x$! All we know is that it is a subset of $U(x)$ and that it is open and nonempty. Thus, a weak basis need not cover the space, so need not be a basis.
For example, consider the topology of the empty set together with the cofinite sets (sets whose complement is finite) on the set of non-negative integers. A weak basis would be the set of cofinite sets of positive integers, but this cannot be a basis, having no neighborhood of $0.$
In general--among $T_1$ spaces, anyway--I suspect that if a space has the property that every weak basis is a basis, then the space is discrete. (The converse trivially holds.)
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
edited Mar 29 at 11:38
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
answered Mar 29 at 11:28
Cameron BuieCameron Buie
86.4k773161
86.4k773161
add a comment |
add a comment |
$begingroup$
A base covers the space.
A weak base may not cover the space.
The set of all not empty, open subsets of R that exclude 0 is a weak base.
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
$endgroup$
add a comment |
$begingroup$
A base covers the space.
A weak base may not cover the space.
The set of all not empty, open subsets of R that exclude 0 is a weak base.
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
$endgroup$
add a comment |
$begingroup$
A base covers the space.
A weak base may not cover the space.
The set of all not empty, open subsets of R that exclude 0 is a weak base.
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
$endgroup$
A base covers the space.
A weak base may not cover the space.
The set of all not empty, open subsets of R that exclude 0 is a weak base.
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
answered Mar 29 at 11:21
William ElliotWilliam Elliot
8,9562820
8,9562820
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%2f3166999%2fis-a-weak-basis-of-a-topological-space-a-basis-and-vice-versa-kechris-book%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
1
$begingroup$
Even a weak basis that is a cover of $X$ need not be a base for $X$. And e.g. $beta omega$ has a countable pseudobase (the singletons of $omega$) but its smallest base has size $2^mathfrakc$.
$endgroup$
– Henno Brandsma
Mar 29 at 22:37