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epimorphism; Urysohn space; cointersection; factorization; natural sink; periodic; cowellpowered; ordinal
Let ${(e_\beta : {\bold Q} \rightarrow Y_\beta)}_{\beta \in \text{\bf Ord}}$ be the large source of epimorphisms in the category $\text{\bf Ury}$ of Urysohn spaces constructed in [2]. A sink ${(g_\beta : Y_\beta \rightarrow X)}_{\beta \in \text{\bf Ord}}$ is called natural, if $g_\beta \circ e_\beta = g_{\beta'} \circ e_{\beta'}$ for all $\beta,\beta' \in \text{\bf Ord}$. In this paper natural sinks are characterized. As a result it is shown that $\text{\bf Ury}$ permits no $({Epi},{\Cal M})$-factorization structure for arbitrary (large) sources.
[1] Adámek J., Herrlich H., Strecker G.E.: Abstract and Concrete Categories. Wiley & Sons 1990. MR 1051419
[2] Schröder J.: The category of Urysohn spaces is not cowellpowered. Top. Appl. 16 (1983), 237-241. MR 0722116
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