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Between countably compact and $\omega$-bounded
General Topology · Submitted 30 Jun 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We give examples in ZFC that separate all these boundedness properties and their appropriate combinations.
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The abstract, sentence by sentence
Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D, \omega N, C_2 \}$ where $\omega D \, \equiv$ "countable discrete", $\omega N \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". Clearly, for each of these $P$-bounded is between countably compact and $\omega$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $\omega D$-bounded spaces which show that $\omega D$-boundedness is much stronger than countable compactness: $\bullet$ Regular $\omega D$-bounded spaces of Lindel\"of degree $< cov(\mathcal{M})$ are $\omega$-bounded. $\bullet$ Regular $\omega D$-bounded spaces of countable tightness are $\omega N$-bounded, and if $\mathfrak{b} > \omega_1$ then even $\omega$-bounded. $\bullet$ If a product of Hausdorff space is $\omega D$-bounded then all but one of its factors must be $\omega$-bounded. $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $\omega D$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. | background 1.00 | 0.14 | 0.02 | asserted 0.97 |
| s1 | Here we study $P$-bounded spaces for the properties $P \in \{\omega D, \omega N, C_2 \}$ where $\omega D \, \equiv$ "countable discrete", $\omega N \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". | contribution 0.80 | 0.74 | 0.04 | asserted 0.97 |
| s2 | Clearly, for each of these $P$-bounded is between countably compact and $\omega$-bounded. | background 0.89 | 0.20 | 0.07 | asserted 1.00 |
| s3 | We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. | contribution 0.75 | 0.96 | 0.33 | asserted 1.00 |
| s4 | Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. | contribution 0.80 | 0.94 | 0.07 | asserted 0.93 |
| s5 | We have interesting results concerning $\omega D$-bounded spaces which show that $\omega D$-boundedness is much stronger than countable compactness: $\bullet$ Regular $\omega D$-bounded spaces of Lindel\"of degree $< cov(\mathcal{M})$ are $\omega$-bounded. | result 0.97 | 0.88 | 0.17 | asserted 1.00 |
| s6 | $\bullet$ Regular $\omega D$-bounded spaces of countable tightness are $\omega N$-bounded, and if $\mathfrak{b} > \omega_1$ then even $\omega$-bounded. | result 1.00 | 0.88 | 0.09 | asserted 1.00 |
| s7 | $\bullet$ If a product of Hausdorff space is $\omega D$-bounded then all but one of its factors must be $\omega$-bounded. | result 1.00 | 0.86 | 0.08 | asserted 1.00 |
| s8 | $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $\omega D$-bounded spaces is countably compact. | result 1.00 | 0.89 | 0.08 | asserted 1.00 |
| s9 | As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated. | result 0.97 | 0.90 | 0.08 | asserted 1.00 |
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Original abstract
Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D, \omega N, C_2 \}$ where $\omega D \, \equiv$ "countable discrete", $\omega N \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". Clearly, for each of these $P$-bounded is between countably compact and $\omega$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $\omega D$-bounded spaces which show that $\omega D$-boundedness is much stronger than countable compactness: $\bullet$ Regular $\omega D$-bounded spaces of Lindel\"of degree $< cov(\mathcal{M})$ are $\omega$-bounded. $\bullet$ Regular $\omega D$-bounded spaces of countable tightness are $\omega N$-bounded, and if $\mathfrak{b} > \omega_1$ then even $\omega$-bounded. $\bullet$ If a product of Hausdorff space is $\omega D$-bounded then all but one of its factors must be $\omega$-bounded. $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $\omega D$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.
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A somewhat similar construction has recently been carried out in [ref] , however not in ZFC but using the extra assumption $𝔭 = (ℳ)$.
Remote Filters and Discretely Generated Spaces · Small and/or locally compact separating examples · 0.99
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