Injections into function spaces over ordinals
For topological spaces \(X\), \(Y\), \(Z\) let \(C_p(X, Z)\) be the space of continuous mappings from \(X\) into \(Z\) with the topology of pointwise convergence. The present paper is one of several recent studies of the relation between spaces \(X\), \(Y\) of ordinals and the presence of some sort of relation between \(C_p(X, Z)\) and \(C_p(Y, Z)\). The case where \(Z\) is a finite cyclic group was studied by \textit{L. V. Genze}, \textit{S. P. Gul'ko} and \textit{T. E. Khmyleva} [Topol. Proc. 38, 1--15 (2011; Zbl 1210.54034)]. The case where \(Z\) is a metric space with two or more points is studied in the present paper. The author shows, among other things, that if \(\tau\) is a regular uncountable cardinal, \(X\subset\tau\) is unbounded, and there is a continuous injection from \(C_p(X, Z)\) into \(C_p(T, Z)\), then \(X\) contains a closed subspace homeomorphic to \(\tau\). Her main result says that if \(X\subset\omega_1\), then the following are equivalent: (i) \(X\) is either countable or contains a homeomorphic copy of \(\omega_1\); (ii) there is a continuous injection from \(C_p(X, Z^\omega)\) into \(C_p(\omega_1, Z^\omega)\). Some open questions are also raised.
scientific article; zbMATH DE number 5835564
scientific article