If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$...
$begingroup$
If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.
Can anyone give me a trivial counter example?
general-topology compactness
asked Dec 6 '18 at 5:45
cmicmi
1,126212
$endgroup$
add a comment |
$begingroup$
If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.
Can anyone give me a trivial counter example?
general-topology compactness
asked Dec 6 '18 at 5:45
cmicmi
1,126212
$endgroup$
1
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59
add a comment |
$begingroup$
If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.
Can anyone give me a trivial counter example?
general-topology compactness
asked Dec 6 '18 at 5:45
cmicmi
1,126212
$endgroup$
If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.
Can anyone give me a trivial counter example?
general-topology compactness
general-topology compactness
asked Dec 6 '18 at 5:45
cmicmi
1,126212
asked Dec 6 '18 at 5:45
cmicmi
1,126212
asked Dec 6 '18 at 5:45
cmicmi
1,126212
asked Dec 6 '18 at 5:45
cmicmi
1,126212
asked Dec 6 '18 at 5:45
cmicmi
1,126212
1,126212
1
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59
add a comment |
1
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59
1
1
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59
add a comment |
1 Answer
1
active
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votes
$begingroup$
Let $Y$ be $omega_1 + 1$ and $X$ is $omega_1$, in the order topology.
Here $omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).
Another example: let $Y$ be $[0,1]^mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Let $Y$ be $omega_1 + 1$ and $X$ is $omega_1$, in the order topology.
Here $omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).
Another example: let $Y$ be $[0,1]^mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
add a comment |
$begingroup$
Let $Y$ be $omega_1 + 1$ and $X$ is $omega_1$, in the order topology.
Here $omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).
Another example: let $Y$ be $[0,1]^mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
add a comment |
$begingroup$
Let $Y$ be $omega_1 + 1$ and $X$ is $omega_1$, in the order topology.
Here $omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).
Another example: let $Y$ be $[0,1]^mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
Let $Y$ be $omega_1 + 1$ and $X$ is $omega_1$, in the order topology.
Here $omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).
Another example: let $Y$ be $[0,1]^mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
edited Dec 6 '18 at 16:40
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
answered Dec 6 '18 at 6:10
Henno BrandsmaHenno Brandsma
107k347114
107k347114
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
add a comment |
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
what is first countable ordinal? Is there no simple counter example?@Henno Brandsma
$endgroup$
– cmi
Dec 6 '18 at 6:20
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
$begingroup$
@cmi Munkres calls it $W$. Look it up
$endgroup$
– Henno Brandsma
Dec 6 '18 at 7:02
add a comment |
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1
$begingroup$
Can you define the term 'limit point compact'?
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 5:51
$begingroup$
If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy
$endgroup$
– cmi
Dec 6 '18 at 5:59