Subspace topology and product topology
$begingroup$
Let $tau$ be a topology on a topological space $X times Y$ which is not a product topology. Consider the subspace topology $tau_X$ and $tau_Y$ induced by the topology $tau$. My question is would the product topology $tau_X times tau_Y$ on $X times Y$ coincide with the topology $tau$.
I tried proving this by showing open sets in one topology is contained in other. I think I can show that $tau subset tau_X times tau_Y$ but I am not able to show the other way round.
general-topology
asked Dec 7 '18 at 15:51
ShreyShrey
132
$endgroup$
add a comment |
$begingroup$
Let $tau$ be a topology on a topological space $X times Y$ which is not a product topology. Consider the subspace topology $tau_X$ and $tau_Y$ induced by the topology $tau$. My question is would the product topology $tau_X times tau_Y$ on $X times Y$ coincide with the topology $tau$.
I tried proving this by showing open sets in one topology is contained in other. I think I can show that $tau subset tau_X times tau_Y$ but I am not able to show the other way round.
general-topology
asked Dec 7 '18 at 15:51
ShreyShrey
132
$endgroup$
2
$begingroup$
$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59
add a comment |
$begingroup$
Let $tau$ be a topology on a topological space $X times Y$ which is not a product topology. Consider the subspace topology $tau_X$ and $tau_Y$ induced by the topology $tau$. My question is would the product topology $tau_X times tau_Y$ on $X times Y$ coincide with the topology $tau$.
I tried proving this by showing open sets in one topology is contained in other. I think I can show that $tau subset tau_X times tau_Y$ but I am not able to show the other way round.
general-topology
asked Dec 7 '18 at 15:51
ShreyShrey
132
$endgroup$
Let $tau$ be a topology on a topological space $X times Y$ which is not a product topology. Consider the subspace topology $tau_X$ and $tau_Y$ induced by the topology $tau$. My question is would the product topology $tau_X times tau_Y$ on $X times Y$ coincide with the topology $tau$.
I tried proving this by showing open sets in one topology is contained in other. I think I can show that $tau subset tau_X times tau_Y$ but I am not able to show the other way round.
general-topology
general-topology
asked Dec 7 '18 at 15:51
ShreyShrey
132
asked Dec 7 '18 at 15:51
ShreyShrey
132
asked Dec 7 '18 at 15:51
ShreyShrey
132
asked Dec 7 '18 at 15:51
ShreyShrey
132
asked Dec 7 '18 at 15:51
ShreyShrey
132
132
2
$begingroup$
$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59
add a comment |
2
$begingroup$
$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59
2
2
$begingroup$
$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59
$begingroup$
$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59
add a comment |
1 Answer
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$begingroup$
You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
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1 Answer
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$begingroup$
You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
$endgroup$
add a comment |
$begingroup$
You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
$endgroup$
add a comment |
$begingroup$
You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
$endgroup$
You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
answered Dec 7 '18 at 15:58
ΑΘΩΑΘΩ
2463
2463
add a comment |
add a comment |
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$tau$ is not a product topology and $tau_Xtimestau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide.
$endgroup$
– drhab
Dec 7 '18 at 15:59