Error in Lang's definition of weak topology?
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On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims
Let $Y$ be a topological space and let $mathscr{F}$ be a family of
mappings $f colon X to Y$ of $X$ into $Y$. Let $mathscr{B}$ be the
family of all subsets of $X$ consisting of the sets $f^{-1}(W)$ where
$W$ is open in $Y$ and $f$ ranges over $mathscr{F}$. Then we leave to
the reader the verification of the following facts:
$mathscr{B}$ is a base for a topology on $X$, i.e. satisfies
conditions B2 and B2.
...
Here B1 and B2 are given on p. 23
B1. Every element of $X$ lies in some set in $mathscr{B}$.
B2. If $B$, $B'$ are in $mathscr{B}$ and $x in B cap B'$ then there exists some $B''$ in $mathscr{B}$ such that $x in B''$ and
$B'' subset B cap B'$.
It seems to me that the $mathscr{B}$ defined in the first quote from Lang need not satisfy the property B2 in the second quote.
For example, take $X=Re^2$ , $Y=Re$, $f(x_1,x_2) = x_1$, $g(x_1,x_2)=x_2$, $mathscr{F} = {f,g}$, $I=(0,1)$, $B=f^{-1}(I)$, $B'=g^{-1}(I)$.
Then $B cap B' = I times I$ but no subset of this set can be an inverse image under either $f$ or $g$ of any subset of $Re$.
Am I right in believing that this is an error in the book?
general-topology
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
$endgroup$
add a comment |
$begingroup$
On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims
Let $Y$ be a topological space and let $mathscr{F}$ be a family of
mappings $f colon X to Y$ of $X$ into $Y$. Let $mathscr{B}$ be the
family of all subsets of $X$ consisting of the sets $f^{-1}(W)$ where
$W$ is open in $Y$ and $f$ ranges over $mathscr{F}$. Then we leave to
the reader the verification of the following facts:
$mathscr{B}$ is a base for a topology on $X$, i.e. satisfies
conditions B2 and B2.
...
Here B1 and B2 are given on p. 23
B1. Every element of $X$ lies in some set in $mathscr{B}$.
B2. If $B$, $B'$ are in $mathscr{B}$ and $x in B cap B'$ then there exists some $B''$ in $mathscr{B}$ such that $x in B''$ and
$B'' subset B cap B'$.
It seems to me that the $mathscr{B}$ defined in the first quote from Lang need not satisfy the property B2 in the second quote.
For example, take $X=Re^2$ , $Y=Re$, $f(x_1,x_2) = x_1$, $g(x_1,x_2)=x_2$, $mathscr{F} = {f,g}$, $I=(0,1)$, $B=f^{-1}(I)$, $B'=g^{-1}(I)$.
Then $B cap B' = I times I$ but no subset of this set can be an inverse image under either $f$ or $g$ of any subset of $Re$.
Am I right in believing that this is an error in the book?
general-topology
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
$endgroup$
1
$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
1
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38
add a comment |
$begingroup$
On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims
Let $Y$ be a topological space and let $mathscr{F}$ be a family of
mappings $f colon X to Y$ of $X$ into $Y$. Let $mathscr{B}$ be the
family of all subsets of $X$ consisting of the sets $f^{-1}(W)$ where
$W$ is open in $Y$ and $f$ ranges over $mathscr{F}$. Then we leave to
the reader the verification of the following facts:
$mathscr{B}$ is a base for a topology on $X$, i.e. satisfies
conditions B2 and B2.
...
Here B1 and B2 are given on p. 23
B1. Every element of $X$ lies in some set in $mathscr{B}$.
B2. If $B$, $B'$ are in $mathscr{B}$ and $x in B cap B'$ then there exists some $B''$ in $mathscr{B}$ such that $x in B''$ and
$B'' subset B cap B'$.
It seems to me that the $mathscr{B}$ defined in the first quote from Lang need not satisfy the property B2 in the second quote.
For example, take $X=Re^2$ , $Y=Re$, $f(x_1,x_2) = x_1$, $g(x_1,x_2)=x_2$, $mathscr{F} = {f,g}$, $I=(0,1)$, $B=f^{-1}(I)$, $B'=g^{-1}(I)$.
Then $B cap B' = I times I$ but no subset of this set can be an inverse image under either $f$ or $g$ of any subset of $Re$.
Am I right in believing that this is an error in the book?
general-topology
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
$endgroup$
On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims
Let $Y$ be a topological space and let $mathscr{F}$ be a family of
mappings $f colon X to Y$ of $X$ into $Y$. Let $mathscr{B}$ be the
family of all subsets of $X$ consisting of the sets $f^{-1}(W)$ where
$W$ is open in $Y$ and $f$ ranges over $mathscr{F}$. Then we leave to
the reader the verification of the following facts:
$mathscr{B}$ is a base for a topology on $X$, i.e. satisfies
conditions B2 and B2.
...
Here B1 and B2 are given on p. 23
B1. Every element of $X$ lies in some set in $mathscr{B}$.
B2. If $B$, $B'$ are in $mathscr{B}$ and $x in B cap B'$ then there exists some $B''$ in $mathscr{B}$ such that $x in B''$ and
$B'' subset B cap B'$.
It seems to me that the $mathscr{B}$ defined in the first quote from Lang need not satisfy the property B2 in the second quote.
For example, take $X=Re^2$ , $Y=Re$, $f(x_1,x_2) = x_1$, $g(x_1,x_2)=x_2$, $mathscr{F} = {f,g}$, $I=(0,1)$, $B=f^{-1}(I)$, $B'=g^{-1}(I)$.
Then $B cap B' = I times I$ but no subset of this set can be an inverse image under either $f$ or $g$ of any subset of $Re$.
Am I right in believing that this is an error in the book?
general-topology
general-topology
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
asked Aug 23 '13 at 16:46
Jyotirmoy BhattacharyaJyotirmoy Bhattacharya
3,14512240
3,14512240
1
$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
1
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38
add a comment |
1
$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
1
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38
1
1
$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
1
1
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38
add a comment |
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This community wiki solution is intended to clear the question from the unanswered queue.
The question has been answered in comments.
You are right, it is a mistake. Lang's $mathscr{B}$ is in general only a subbase.
$endgroup$
add a comment |
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$begingroup$
This community wiki solution is intended to clear the question from the unanswered queue.
The question has been answered in comments.
You are right, it is a mistake. Lang's $mathscr{B}$ is in general only a subbase.
$endgroup$
add a comment |
$begingroup$
This community wiki solution is intended to clear the question from the unanswered queue.
The question has been answered in comments.
You are right, it is a mistake. Lang's $mathscr{B}$ is in general only a subbase.
$endgroup$
add a comment |
$begingroup$
This community wiki solution is intended to clear the question from the unanswered queue.
The question has been answered in comments.
You are right, it is a mistake. Lang's $mathscr{B}$ is in general only a subbase.
$endgroup$
This community wiki solution is intended to clear the question from the unanswered queue.
The question has been answered in comments.
You are right, it is a mistake. Lang's $mathscr{B}$ is in general only a subbase.
edited Dec 23 '18 at 16:42
community wiki
2 revs, 2 users 89%
Paul Frost
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$begingroup$
The base of the weak topology consists of all finite intersections of sets like $f^{-1}(W)$, as $f$ ranges in $mathcal{F}$ and $W$ over the open subsets of $Y$. It seems to me that Lang's $mathcal{B}$ is only a sub-base.
$endgroup$
– Siminore
Aug 23 '13 at 17:11
1
$begingroup$
Right, it's a mistake. ${ f^{-1}(W) : f in mathscr{F}, W subset Y text{ open}}$ is only a subbase.
$endgroup$
– Daniel Fischer♦
Aug 23 '13 at 17:38