## Abstract

Let be a property (or, equivalently, a class) of topological spaces. A space *X* is called -**bounded** if every subspace of *X* with (or in) has compact closure. Thus, countable-bounded has been known as *ω*-bounded and (*σ*-compact)-bounded as strongly *ω*-bounded.

In this paper we present a systematic study of the interrelations of these two known “boundedness” concepts with -boundedness where is one of the further countability properties weakly Lindelöf, Lindelöf, hereditarily Lindelöf, and *ccc*.

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## Additional information

The first author was supported by OTKA grants no. 68262 and 83726.

The second and third author are pleased to thank the Alfréd Rényi Institute of Mathematics for generous hospitality.

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Juhász, I., van Mill, J. & Weiss, W. Variations on *ω*-boundedness.
*Isr. J. Math.* **194**, 745–766 (2013). https://doi.org/10.1007/s11856-012-0062-8

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DOI: https://doi.org/10.1007/s11856-012-0062-8

### Keywords

- Compact Space
- Continuum Hypothesis
- Nonempty Open Subset
- Compact Closure
- Clopen Subset