Definition
In topology, a topological space $X$ is called ω‑bounded (read “omega‑bounded”) if the closure of every countable subset of $X$ is compact. Equivalently, each countable subset of $X$ has a compact closure, or every countable subset is relatively compact in $X$.
Relation to other compactness properties
| Property | Definition | Implication |
|---|---|---|
| Compact | Every open cover has a finite subcover. | ⇒ ω‑bounded |
| ω‑bounded | Closure of each countable set is compact. | ⇒ Countably compact |
| Countably compact | Every countable open cover has a finite subcover (or every infinite subset has an accumulation point). | — |
| Pseudocompact | Every real‑valued continuous function on $X$ is bounded. | — |
Thus:
$$
\text{Compact} ;\Longrightarrow; \omega\text{-bounded} ;\Longrightarrow; \text{Countably compact},
$$
and none of the reverse implications hold in general.
Typical examples
- The first uncountable ordinal $\omega_{1}$ with the order topology is ω‑bounded but not compact. Every countable subset has a supremum less than $\omega_{1}$; its closure is a countable ordinal $\alpha+1$, which is compact.
- The product $\omega_{1}+1 \times [0,1]$ (with the product topology) is ω‑bounded and not compact, illustrating that ω‑boundedness can be preserved under certain product constructions.
- Stone–Čech compactification of $\mathbb{N}$ minus $\mathbb{N}$, denoted $\beta\mathbb{N}\setminus\mathbb{N}$, is a classic ω‑bounded space that is not compact.
Non‑examples
- The countable discrete space $\mathbb{N}$ is not ω‑bounded, because the closure of a countable subset (itself) is not compact.
- The Sorgenfrey line (real line with lower‑limit topology) is not ω‑bounded; it contains countable subsets whose closures fail to be compact.
Key properties
- Preservation under continuous images: If $f:X\to Y$ is continuous and $X$ is ω‑bounded, then $f[X]$ is ω‑bounded.
- Closed subspaces: Any closed subspace of an ω‑bounded space is ω‑bounded.
- Products: Finite products of ω‑bounded spaces are ω‑bounded; arbitrary products need not be ω‑bounded (e.g., $\prod_{i\in I}\omega_{1}$ for uncountable $I$ is not ω‑bounded).
- Characterization via nets: A space is ω‑bounded iff every countable net has a convergent subnet.
Historical notes
The notion of ω‑boundedness was introduced in the mid‑20th century within the study of generalized compactness properties. It appears in the works of H. J. P. M. de Bruijn and J. van Mill, and was later systematized in textbooks on general topology (e.g., Engelking, General Topology, 1989).
References
- Engelking, R. General Topology. Sigma Series in Pure Mathematics, 1989.
- van Mill, J. The Infinite-Dimensional Topology of Function Spaces. North-Holland, 2001.
- M. R. Kelley, General Topology. Springer, 1975.
These sources provide detailed discussions of ω‑bounded spaces, their properties, and examples.