Topological spaces with prescribed nonconstant continuous mappings

Trnková, Věra

doi:10.1090/S0002-9947-1980-0580898-9

Topological spaces with prescribed nonconstant continuous mappings
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Trans. Amer. Math. Soc. 261 (1980), 463-482 Request permission

Abstract:

Given a ${T_1}$-space Y and a ${T_3}$-space V, consider ${T_3}$-spaces X such that X has a closed covering by spaces homeomorphic to V and any continuous mapping $f: X \to Y$ is constant. All such spaces and all their continuous mappings are shown to form a very comprehensive category, containing, e.g., a proper class of spaces without nonconstant, nonidentical mappings or containing a space X, for every monoid M, such that all the nonconstant continuous mappings of X into itself are closed under composition and form a monoid isomorphic to M. The category of paracompact connected spaces, having a closed covering by a given totally disconnected paracompact space, has, e.g., analogous properties. Categories of metrizable spaces are also investigated.

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