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Natural sinks on \(Y_ \beta\). (English) Zbl 0761.18004
For the rationals \(\mathbb{Q}\) with the natural topology and an ordinal \(\beta\), the author constructed a Urysohn space \(Y_ \beta\) and an epimorphism \(e_ \beta: \mathbb{Q}\to Y_ \beta\) in \(\mathbf{Ury}\), the category of Urysohn spaces, and showed that the category \(\mathbf{Ury}\) is not cowellpowered [Topology Appl. 16, 237-241 (1983; Zbl 0534.54004)]. In this paper, he gives a characterization of a sink \((g_ \beta: Y_ \beta\to X)_{\beta\in\text{Ord}}\) satisfying \(g_ \beta\circ e_ \beta=g_{\beta'}\circ e_{\beta'}\) for all \(\beta,\beta'\in\text{Ord}\) and shows that a large source \((e_ \beta: Q\to Y_ \beta)_{\beta\in\text{Ord}}\) has no cointersection in \(\mathbf{Ury}\). This implies that \(\mathbf{Ury}\) permits no \((Epi,{\mathcal M})\)-factorization structure for arbitrary sources.
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54B30 Categorical methods in general topology
Zbl 0534.54004
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