Openly factorizable spaces and compact extensions of topological semigroups

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-\v{C}ech compactification $\beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

💡 Research Summary

The paper investigates the long‑standing problem of extending the binary operation of a topological semigroup to its Stone‑Čech compactification. While it is well known that compact semigroups or those whose operation is already jointly continuous admit such an extension, the situation for non‑compact, merely pseudocompact semigroups has remained obscure. The authors introduce a new class of spaces—openly factorizable spaces—to fill this gap.

A space X is called openly factorizable if for every continuous map f : X → Y into a second‑countable space Y there exist an open surjection p : X → Z onto a second‑countable space Z and a continuous map g : Z → Y such that f = g ∘ p. This condition strengthens the classical notion of factorizability by requiring the intermediate map to be open, which gives better control over the topology of images and pre‑images. The authors first establish several permanence properties: the class is closed under taking subspaces, products, and continuous images, and every metrizable space belongs to it, whereas many non‑metrizable compact spaces do not.

The central theorem states: If S is a pseudocompact, openly factorizable topological semigroup, then the semigroup operation * : S × S → S extends to a continuous operation on the Stone‑Čech compactification βS. The proof proceeds by exploiting the factorization property for arbitrary continuous real‑valued functions on S. Given such a function f, one writes f = g ∘ p with p open and g continuous. Because p is open, its Stone‑Čech extension βp : βS → βZ is continuous, where Z is second‑countable and therefore βZ is a metrizable compact space. The continuity of βp allows the authors to transfer the joint continuity of the original operation to βS via the diagram involving βp and the continuous map βg. In effect, the operation on βS is defined by β*(x, y) = βg(βp(x)·βp(y)), and the joint continuity follows from the continuity of βp, βg, and the multiplication on the metrizable compact βZ.

Beyond the main extension theorem, the paper provides a spectral characterization of openly factorizable spaces. The authors construct an “inverse‑image spectrum” consisting of all open surjections onto second‑countable spaces and show that a space is openly factorizable precisely when it can be recovered as the inverse limit of this spectrum. This spectral viewpoint yields a clear structural picture: openly factorizable spaces are exactly those whose topology can be assembled from countable, metrizable pieces glued together via open bonding maps.

Several illustrative examples are given. All metrizable spaces, including the real line, Hilbert cube, and countable products of intervals, satisfy the definition. In contrast, the remainder βℕ \ ℕ (the Čech–Stone remainder of the natural numbers) fails to be openly factorizable, showing that the condition is non‑trivial. The authors also discuss how the class interacts with well‑known compactifications: for an openly factorizable space X, the Hewitt realcompactification νX coincides with the closure of X in its Stone‑Čech compactification, reinforcing the tight link between openness, factorization, and compact extensions.

The paper concludes with a list of open problems. One direction asks whether every pseudocompact semitopological semigroup can be re‑topologized to become openly factorizable without altering its algebraic structure. Another concerns the uniqueness of the extended operation on βS: does the extension preserve additional algebraic properties such as idempotency, cancellativity, or the existence of minimal ideals? Finally, the authors suggest exploring connections with other factorization concepts in functional analysis, such as the factorization of operators through Banach spaces with the approximation property, to see whether analogous “open” versions can be defined.

In summary, the article makes a substantial contribution to the theory of topological semigroups by identifying openly factorizable spaces as the natural setting where the semigroup operation extends continuously to the Stone‑Čech compactification. The spectral characterization deepens our understanding of these spaces, and the results open several promising avenues for further research in both pure topology and topological algebra.