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If $H$ and $G/H$ are pseudocompact then is G also pseudocompact?
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$begingroup$
Theorem 5.25 in Edwin, Hewitt Abstract Harmonic Analysis Part 1, says if $H$ is a subgroup of a topological group $G$ and $H$ and $G/H$ are compact (resp. locally compact), then so is $G$.
For my purpose I can assume topological groups to be Hausdorff, if necessary. My question is does this property extends to weaker forms of compactness, like pseudocompact or quasicompact (if every covering of G by co-zero sets admits a finite subcovering). Is such a result already known to be true or false? If not which way should I look for? Any help is appreciated.
general-topology topological-groups
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
$endgroup$
add a comment |
$begingroup$
Theorem 5.25 in Edwin, Hewitt Abstract Harmonic Analysis Part 1, says if $H$ is a subgroup of a topological group $G$ and $H$ and $G/H$ are compact (resp. locally compact), then so is $G$.
For my purpose I can assume topological groups to be Hausdorff, if necessary. My question is does this property extends to weaker forms of compactness, like pseudocompact or quasicompact (if every covering of G by co-zero sets admits a finite subcovering). Is such a result already known to be true or false? If not which way should I look for? Any help is appreciated.
general-topology topological-groups
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
$endgroup$
add a comment |
$begingroup$
Theorem 5.25 in Edwin, Hewitt Abstract Harmonic Analysis Part 1, says if $H$ is a subgroup of a topological group $G$ and $H$ and $G/H$ are compact (resp. locally compact), then so is $G$.
For my purpose I can assume topological groups to be Hausdorff, if necessary. My question is does this property extends to weaker forms of compactness, like pseudocompact or quasicompact (if every covering of G by co-zero sets admits a finite subcovering). Is such a result already known to be true or false? If not which way should I look for? Any help is appreciated.
general-topology topological-groups
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
$endgroup$
Theorem 5.25 in Edwin, Hewitt Abstract Harmonic Analysis Part 1, says if $H$ is a subgroup of a topological group $G$ and $H$ and $G/H$ are compact (resp. locally compact), then so is $G$.
For my purpose I can assume topological groups to be Hausdorff, if necessary. My question is does this property extends to weaker forms of compactness, like pseudocompact or quasicompact (if every covering of G by co-zero sets admits a finite subcovering). Is such a result already known to be true or false? If not which way should I look for? Any help is appreciated.
general-topology topological-groups
general-topology topological-groups
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
asked Mar 16 at 7:04
Bhaskar VashishthBhaskar Vashishth
7,83712055
7,83712055
add a comment |
add a comment |
1 Answer
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$begingroup$
It is well-known that the product of two pseudocompact topological groups is again pseudocompact (this need not hold for general spaces, but does hold in Hausdorff topological groups), and there is a map from $H times G/{H}$ onto $G$ I believe and this would imply your question for the pseudocompact case (isn't this also how the Hewitt proof for (local) compactness goes?)
Quasicompactness and compactness are just equivalent for Hausdorff topological groups (or in general for $T_{3frac12}$ spaces). So if you can assume Hausdorffness (or $T_0$ which is equivalent) for your group, there's nothing new there.
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
$endgroup$
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
add a comment |
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1 Answer
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$begingroup$
It is well-known that the product of two pseudocompact topological groups is again pseudocompact (this need not hold for general spaces, but does hold in Hausdorff topological groups), and there is a map from $H times G/{H}$ onto $G$ I believe and this would imply your question for the pseudocompact case (isn't this also how the Hewitt proof for (local) compactness goes?)
Quasicompactness and compactness are just equivalent for Hausdorff topological groups (or in general for $T_{3frac12}$ spaces). So if you can assume Hausdorffness (or $T_0$ which is equivalent) for your group, there's nothing new there.
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
$endgroup$
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
add a comment |
$begingroup$
It is well-known that the product of two pseudocompact topological groups is again pseudocompact (this need not hold for general spaces, but does hold in Hausdorff topological groups), and there is a map from $H times G/{H}$ onto $G$ I believe and this would imply your question for the pseudocompact case (isn't this also how the Hewitt proof for (local) compactness goes?)
Quasicompactness and compactness are just equivalent for Hausdorff topological groups (or in general for $T_{3frac12}$ spaces). So if you can assume Hausdorffness (or $T_0$ which is equivalent) for your group, there's nothing new there.
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
$endgroup$
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
add a comment |
$begingroup$
It is well-known that the product of two pseudocompact topological groups is again pseudocompact (this need not hold for general spaces, but does hold in Hausdorff topological groups), and there is a map from $H times G/{H}$ onto $G$ I believe and this would imply your question for the pseudocompact case (isn't this also how the Hewitt proof for (local) compactness goes?)
Quasicompactness and compactness are just equivalent for Hausdorff topological groups (or in general for $T_{3frac12}$ spaces). So if you can assume Hausdorffness (or $T_0$ which is equivalent) for your group, there's nothing new there.
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
$endgroup$
It is well-known that the product of two pseudocompact topological groups is again pseudocompact (this need not hold for general spaces, but does hold in Hausdorff topological groups), and there is a map from $H times G/{H}$ onto $G$ I believe and this would imply your question for the pseudocompact case (isn't this also how the Hewitt proof for (local) compactness goes?)
Quasicompactness and compactness are just equivalent for Hausdorff topological groups (or in general for $T_{3frac12}$ spaces). So if you can assume Hausdorffness (or $T_0$ which is equivalent) for your group, there's nothing new there.
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
answered Mar 16 at 7:21
Henno BrandsmaHenno Brandsma
114k348123
114k348123
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
add a comment |
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
Where can I find a proof that quasicompactness is equivalent to compactness for Hausdorff topological groups?
$endgroup$
– Bhaskar Vashishth
Mar 16 at 7:50
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
$begingroup$
@BhaskarVashishth it’s trivial because in a Tychonoff space the co-zero sets are a base.
$endgroup$
– Henno Brandsma
Mar 16 at 7:51
add a comment |
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