Subspaces of pseudoradial spaces

We prove that every topological space (T0-space, T1-space) can be embedded in a pseudoradial space (in a pseudoradial T0-space, T1space). This answers the Problem 3 in [Arhangelskii, A.V. - Isler, R. - Tironi, G: On pseudo-radial spaces, Comment. Math. Univ. Carolin. 27 (1986), 137-156]. We describe the smallest coreflective subcategory A of Top such that the hereditary coreflective hull of A is the whole category Top. (The same result without any separation axiom was proved already in the master thesis E. Murtinov'a: Podprostory pseudoradi'aln'ich prostor\r{u}. I wasn’t aware of this work when preparing this paper.)

💡 Research Summary

The paper addresses two intertwined problems in general topology and categorical topology. The first problem, originally posed by Arhangel’skii, Isler, and Tiroi, asks whether every topological space (and, more specifically, every T₀‑space or T₁‑space) can be realized as a subspace of a pseudoradial space. The second problem seeks the smallest coreflective subcategory A of the category Top such that the hereditary coreflective hull SA (the class of all subspaces of spaces from A) coincides with the whole category Top.

A pseudoradial space is defined by the property that a set is closed whenever it contains the limits of all transfinite sequences it contains. The authors also consider β‑sequential spaces, where the same closure condition is required only for sequences indexed by ordinals ≤ β. They recall that β‑sequential ⇒ pseudoradial and that β‑sequential spaces form a coreflective subcategory Psrad(β) of Top.

The key technical observation is that for any infinite cardinal β, the β‑fold power S^β of the Sierpiński two‑point space S = {0,1} is β‑sequential (Theorem 3.2). The proof proceeds by assuming the contrary, choosing a minimal counterexample, and deriving a contradiction via a carefully constructed net of characteristic functions. Consequently S^β is also 2β‑sequential.

Using this fact, the authors show that any prime space P (a T₀‑space with exactly one accumulation point) of cardinality ≤ α embeds into S^β where β = w(P) ≤ 2α (Theorem 3.3). Since Psrad(2α) is coreflective, every prime space of size ≤ α belongs to SPsrad(2α). Because the coreflective hull of all prime spaces of size ≤ α is precisely Gen(α) (the class of spaces of tightness ≤ α), it follows that Gen(α) ⊆ SPsrad(2α) for every infinite α (Theorem 3.3). Hence any topological space X, being a subspace of some Gen(α), is a subspace of a pseudoradial space (Theorem 3.4).

For T₀‑spaces the embedding is immediate: every T₀‑space embeds into some S^α (a known result). For T₁‑spaces the authors introduce (S^α)_1, the join of the product topology on S^α with the cofinite topology on the same underlying set. They prove that (S^α)_1 is α‑sequential (Proposition 3.5) and therefore pseudoradial. Since any T₁‑space embeds into S^α, it also embeds into (S^α)_1, yielding the T₁ version of the main theorem (Theorem 3.6).

The second part of the paper shifts to a categorical perspective. Let A be a coreflective subcategory of Top. The hereditary coreflective hull SA consists of all subspaces of spaces from A. The authors prove (Theorem 4.1) that SA = Top iff S^α ∈ A for every infinite cardinal α. The proof uses the fact that if S^α is a subspace of some X ∈ A, then the coordinate projections p_a : S^α → S extend to continuous maps on X, yielding a retraction from X onto S^α; thus S^α itself must belong to A. Conversely, if all S^α belong to A, then any prime space (hence any space) is a subspace of a member of A, so SA = Top.

Consequently, the coreflective hull of the class {S^α | α ∈ Cn} (where Cn denotes all infinite cardinals) is the smallest coreflective subcategory with SA = Top (Corollary 4.3). The authors also introduce a simpler generating family M(α), a space on α ∪ {α} with a topology generated by the “tails” B_β = {γ ≥ β}. They show that CH(M(α)) = CH(S^α) (Corollary 4.6) and that the inclusion of M(α) in A is equivalent to the inclusion of S^α (Theorem 4.5). Moreover, they prove (Theorem 4.8) that SA = Top iff M(α) ∈ A for every regular cardinal α.

Finally, the paper characterizes those coreflective subcategories whose hereditary kernel is FG (the class of finitely generated spaces). Theorem 4.9 states that SA = Top iff for each regular cardinal α there exists a space X ∈ A with a point of tightness α, and for α = ω the corresponding prime factor is not finitely generated. This condition guarantees that PsRad ⊆ SA, and by the earlier embedding theorem, SA = Top.

In summary, the authors provide a constructive embedding of any space into a pseudoradial space (including the T₀ and T₁ variants) via powers of the Sierpiński space, and they identify the minimal coreflective subcategory—namely the coreflective hull of the family {S^α}—that suffices to generate all topological spaces through subspace formation. This resolves Problem 3 from the cited literature and enriches the understanding of the interplay between sequential properties, pseudoradiality, and categorical coreflection in topology.