normal doesn't imply paracompact
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I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.
Thanks in advance.
general-topology examples-counterexamples separation-axioms paracompactness
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
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add a comment |
$begingroup$
I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.
Thanks in advance.
general-topology examples-counterexamples separation-axioms paracompactness
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
$endgroup$
add a comment |
$begingroup$
I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.
Thanks in advance.
general-topology examples-counterexamples separation-axioms paracompactness
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
$endgroup$
I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.
Thanks in advance.
general-topology examples-counterexamples separation-axioms paracompactness
general-topology examples-counterexamples separation-axioms paracompactness
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
edited Dec 9 '18 at 3:23
Eric Wofsey
183k13211338
183k13211338
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
asked Jan 22 '18 at 17:32
Chun GanChun Gan
31218
31218
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.
The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).
Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
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add a comment |
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An example is built here, have you read it:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
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2 Answers
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2 Answers
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$begingroup$
The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.
The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).
Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
add a comment |
$begingroup$
The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.
The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).
Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
add a comment |
$begingroup$
The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.
The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).
Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
$endgroup$
The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.
The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).
Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
edited Jan 22 '18 at 22:34
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
answered Jan 22 '18 at 20:13
Henno BrandsmaHenno Brandsma
107k347114
107k347114
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add a comment |
$begingroup$
An example is built here, have you read it:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
$endgroup$
add a comment |
$begingroup$
An example is built here, have you read it:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
$endgroup$
add a comment |
$begingroup$
An example is built here, have you read it:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
$endgroup$
An example is built here, have you read it:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
answered Jan 22 '18 at 18:06
Arnaud MortierArnaud Mortier
19.8k22260
19.8k22260
add a comment |
add a comment |
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