Mixing
countably compact and compact
$begingroup$
I know that compact space is countably compact, and every compact is pseudo-compact.
Is there a simple example that show any Compact space is not necessarily compact
?
Is there an example to show that any pseudo-compact space is not necessarily compact?
general-topology compactness topological-groups
asked Jan 13 at 17:07
adinadin
142
$endgroup$
add a comment |
$begingroup$
I know that compact space is countably compact, and every compact is pseudo-compact.
Is there a simple example that show any Compact space is not necessarily compact
?
Is there an example to show that any pseudo-compact space is not necessarily compact?
general-topology compactness topological-groups
asked Jan 13 at 17:07
adinadin
142
$endgroup$
$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
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Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18
add a comment |
$begingroup$
I know that compact space is countably compact, and every compact is pseudo-compact.
Is there a simple example that show any Compact space is not necessarily compact
?
Is there an example to show that any pseudo-compact space is not necessarily compact?
general-topology compactness topological-groups
asked Jan 13 at 17:07
adinadin
142
$endgroup$
I know that compact space is countably compact, and every compact is pseudo-compact.
Is there a simple example that show any Compact space is not necessarily compact
?
Is there an example to show that any pseudo-compact space is not necessarily compact?
general-topology compactness topological-groups
general-topology compactness topological-groups
asked Jan 13 at 17:07
adinadin
142
asked Jan 13 at 17:07
adinadin
142
edited Jan 13 at 18:47
adin
asked Jan 13 at 17:07
adinadin
142
asked Jan 13 at 17:07
adinadin
142
asked Jan 13 at 17:07
adinadin
142
142
$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
$begingroup$
Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18
add a comment |
$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
$begingroup$
Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18
$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
$begingroup$
Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18
add a comment |
1 Answer
1
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votes
$begingroup$
Okay, so, your initial statement is incorrect. The correct implications are as follows:
$$text{compact};;implies;;text{countably compact};;implies;;text{pseudocompact}$$
(Note that "countably compact" is not the same as "$sigma$-compact.")
To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.
answered Jan 13 at 17:20
Ben WBen W
2,283615
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$begingroup$
Okay, so, your initial statement is incorrect. The correct implications are as follows:
$$text{compact};;implies;;text{countably compact};;implies;;text{pseudocompact}$$
(Note that "countably compact" is not the same as "$sigma$-compact.")
To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.
answered Jan 13 at 17:20
Ben WBen W
2,283615
$endgroup$
add a comment |
$begingroup$
Okay, so, your initial statement is incorrect. The correct implications are as follows:
$$text{compact};;implies;;text{countably compact};;implies;;text{pseudocompact}$$
(Note that "countably compact" is not the same as "$sigma$-compact.")
To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.
answered Jan 13 at 17:20
Ben WBen W
2,283615
$endgroup$
add a comment |
$begingroup$
Okay, so, your initial statement is incorrect. The correct implications are as follows:
$$text{compact};;implies;;text{countably compact};;implies;;text{pseudocompact}$$
(Note that "countably compact" is not the same as "$sigma$-compact.")
To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.
answered Jan 13 at 17:20
Ben WBen W
2,283615
$endgroup$
Okay, so, your initial statement is incorrect. The correct implications are as follows:
$$text{compact};;implies;;text{countably compact};;implies;;text{pseudocompact}$$
(Note that "countably compact" is not the same as "$sigma$-compact.")
To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.
answered Jan 13 at 17:20
Ben WBen W
2,283615
answered Jan 13 at 17:20
Ben WBen W
2,283615
answered Jan 13 at 17:20
Ben WBen W
2,283615
answered Jan 13 at 17:20
Ben WBen W
2,283615
2,283615
add a comment |
add a comment |
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$begingroup$
What does it mean for a space to be compressed?
$endgroup$
– Aweygan
Jan 13 at 17:10
$begingroup$
Excuse me, compact.I edited it.
$endgroup$
– adin
Jan 13 at 17:14
$begingroup$
two more comments: in your first sentence, did you mean to write "and every compact space is pseudo-compact", and in your first question, the word compressed still appears. Should it instead be "Is there a simple example to show that a countably compact space is not compact"?
$endgroup$
– Aweygan
Jan 13 at 17:18