Are countably compactly generated spaces paracompact?
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A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.
Are countably compactly generated spaces paracompact spaces? Do we have partition of unity for countably compactly generated spaces?
general-topology compactness paracompactness
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
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add a comment |
$begingroup$
A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.
Are countably compactly generated spaces paracompact spaces? Do we have partition of unity for countably compactly generated spaces?
general-topology compactness paracompactness
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
$endgroup$
add a comment |
$begingroup$
A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.
Are countably compactly generated spaces paracompact spaces? Do we have partition of unity for countably compactly generated spaces?
general-topology compactness paracompactness
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
$endgroup$
A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.
Are countably compactly generated spaces paracompact spaces? Do we have partition of unity for countably compactly generated spaces?
general-topology compactness paracompactness
general-topology compactness paracompactness
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
edited Dec 9 '18 at 21:57
Eric Wofsey
186k14215342
186k14215342
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
asked Dec 2 '14 at 8:15
Kamran sharifiKamran sharifi
161
161
add a comment |
add a comment |
1 Answer
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It seems that each countably compactly generated space $X$ is $sigma$-compact; so if $X$ is regular then $X$ is Lindelof and, therefore, paracompact.
Moreover, it seems that a Hausdorff space is countably compactly generated iff it is a $k_omega$-space. I remind that a topological space $X$ is defined to be a $k_omega$-space if there is a countable cover $mathcal K$ of $X$ by compact subsets of $X$, determining the topology of $X$ in the sense that a subset $U$ of $X$ is open in $X$ if and only if the intersection $Ucap K$ is open in $K$ for any compact set $Kinmathcal K$. According to [FT], each Hausdorff $k_omega$-space is normal.
[FT] S.P. Franklin, B.V. Smith Thomas. A survey of $k_omega$-spaces // Topology Proc. 2 (1977),
111--124.
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
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1 Answer
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1 Answer
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$begingroup$
It seems that each countably compactly generated space $X$ is $sigma$-compact; so if $X$ is regular then $X$ is Lindelof and, therefore, paracompact.
Moreover, it seems that a Hausdorff space is countably compactly generated iff it is a $k_omega$-space. I remind that a topological space $X$ is defined to be a $k_omega$-space if there is a countable cover $mathcal K$ of $X$ by compact subsets of $X$, determining the topology of $X$ in the sense that a subset $U$ of $X$ is open in $X$ if and only if the intersection $Ucap K$ is open in $K$ for any compact set $Kinmathcal K$. According to [FT], each Hausdorff $k_omega$-space is normal.
[FT] S.P. Franklin, B.V. Smith Thomas. A survey of $k_omega$-spaces // Topology Proc. 2 (1977),
111--124.
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
$endgroup$
add a comment |
$begingroup$
It seems that each countably compactly generated space $X$ is $sigma$-compact; so if $X$ is regular then $X$ is Lindelof and, therefore, paracompact.
Moreover, it seems that a Hausdorff space is countably compactly generated iff it is a $k_omega$-space. I remind that a topological space $X$ is defined to be a $k_omega$-space if there is a countable cover $mathcal K$ of $X$ by compact subsets of $X$, determining the topology of $X$ in the sense that a subset $U$ of $X$ is open in $X$ if and only if the intersection $Ucap K$ is open in $K$ for any compact set $Kinmathcal K$. According to [FT], each Hausdorff $k_omega$-space is normal.
[FT] S.P. Franklin, B.V. Smith Thomas. A survey of $k_omega$-spaces // Topology Proc. 2 (1977),
111--124.
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
$endgroup$
add a comment |
$begingroup$
It seems that each countably compactly generated space $X$ is $sigma$-compact; so if $X$ is regular then $X$ is Lindelof and, therefore, paracompact.
Moreover, it seems that a Hausdorff space is countably compactly generated iff it is a $k_omega$-space. I remind that a topological space $X$ is defined to be a $k_omega$-space if there is a countable cover $mathcal K$ of $X$ by compact subsets of $X$, determining the topology of $X$ in the sense that a subset $U$ of $X$ is open in $X$ if and only if the intersection $Ucap K$ is open in $K$ for any compact set $Kinmathcal K$. According to [FT], each Hausdorff $k_omega$-space is normal.
[FT] S.P. Franklin, B.V. Smith Thomas. A survey of $k_omega$-spaces // Topology Proc. 2 (1977),
111--124.
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
$endgroup$
It seems that each countably compactly generated space $X$ is $sigma$-compact; so if $X$ is regular then $X$ is Lindelof and, therefore, paracompact.
Moreover, it seems that a Hausdorff space is countably compactly generated iff it is a $k_omega$-space. I remind that a topological space $X$ is defined to be a $k_omega$-space if there is a countable cover $mathcal K$ of $X$ by compact subsets of $X$, determining the topology of $X$ in the sense that a subset $U$ of $X$ is open in $X$ if and only if the intersection $Ucap K$ is open in $K$ for any compact set $Kinmathcal K$. According to [FT], each Hausdorff $k_omega$-space is normal.
[FT] S.P. Franklin, B.V. Smith Thomas. A survey of $k_omega$-spaces // Topology Proc. 2 (1977),
111--124.
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
edited Dec 8 '14 at 9:49
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
answered Dec 8 '14 at 5:09
Alex RavskyAlex Ravsky
41.4k32282
41.4k32282
add a comment |
add a comment |
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