What are some nice characterizations of $mathfrak{c}$-compact spaces?
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We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
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add a comment |
$begingroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
add a comment |
$begingroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $kappa$-sized subcover, for any infinite cardinal $kappa$.
I imagine that $kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $mathfrak{c}$-compact spaces?
general-topology
general-topology
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 18 '18 at 9:14
MathematicsStudent1122MathematicsStudent1122
8,98332668
8,98332668
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
add a comment |
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18
add a comment |
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$begingroup$
The definition is the characterisation.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:12
1
$begingroup$
BTW it’s usual to define $X$ to be $kappa$-compact when every open cover of $X$ has a subcover of size $<kappa$. So you ask about $mathfrak{c}^+$-compact spaces. $omega$-compact is usual compact and Lindelöf is $aleph_1$-compact. The compactness number $l(X)$ is defined as the smallest infinite cardinal $kappa$ such that $X$ is $kappa$-compact.
$endgroup$
– Henno Brandsma
Dec 18 '18 at 12:18