On the compact open and finest splitting topologies (Q886304)

From the authors' summary: In [\textit{M. H. Escardo, J. Lawson} and \textit{A. Simpson}, Topology Appl. 143, No. 1--3, 105--145 (2004; Zbl 1066.54028)] it was shown that in the set \(C(N^{\omega },N)\) of all continuous maps of \(N^{\omega }\) into \(N\), where \(N\) is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since \(N^{\omega }\) is consonant [see \textit{S. Dolecki, G. H. Greco} and \textit{A. Lechicki}, Trans. Am. Math. Soc. 347, No. 8, 2869--2884 (1995; Zbl 0845.54005)], the Isbell topology on \(C(N^{\omega },N)\) is not the finest splitting topology, either. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property.'' Some of the used notations and definitions are as follows: Let \(Y\) and \(Z\) be two spaces and \(C(Y,Z)\) the collection of all continuous maps of \(Y\) into \(Z\). Then the topology \(t\) on \(C(Y,Z)\) is called splitting if for every space \(X\), the continuity of a map \(g: X\times Y\to Z\) implies that of the map \(\widehat g: X\to C_t(Y, Z)\), where \(\widehat g: Y\to Z\) is defined by \(g_x(y)= g(x, y)\) for \(x\in X\) and \(y\in Y\), and \(\widehat g(x)= g_x\) for every \(x\in X\). A space \(M\) has the specific-extension property if it satisfies the following conditions (a) there exists a subset \(\{z_0,z_1,\dots\}\) of distinct points of \(M\) and (b) for every compact subset \(K\) of \(M^\omega\) and an element \(\psi\) of \(C(M^\omega, M)\) there exist an integer \(m\geq 0\) and a point \(y\equiv(y_0,y_1,\dots)\in M^\omega\) such that: (1) \(\psi(\overline{z_i})= y_i, i\in \{0,\dots, m-1\}\), \((\{0,\dots, m-1\}= \emptyset\) if \(m= 0)\). (2) \(y\not\in K\cup\{\overline{z_0}, \overline{z_1},\dots\}\), \(\overline{z_i}\not\in K\) if \(i\geq m\), and the singletons \(\{y\}\) and \(\{\overline{z_i}\}\), \(i\geq m\), are simultaneously open and closed subsets of \(K\cup\{\overline{z_0}, \{\overline{z_1},\dots\}\cup \{y\}\) (in the relative topology). (3) Each continuous map \(\varphi: K\cup\{\overline z_0,\overline z_1,\dots\}\cup \{y\}\to M\) for which \(\varphi|K\cup\{\overline z_0,\dots,\overline z_{m-1}\}= \psi|K\cup\{\overline z_0,\dots,\overline z_{m-1}\}\) has a continuous extension \(f: M^\omega\to M\). A space having the specific extension property is called an SEP-space. The main results of the paper under review are as below: Theorem 3.4. If \(M\) is an SEP-space, then the compact-open topology on \(C(M^\omega, M)\) does not coincide with the finest splitting topology. Theorem 3.6. An infinitely countable free union \(P\) of non-empty spaces is an SEP-space and, therefore, the compact-open topology on \(C(P^\omega, P)\) does not coincide with the finest splitting topology.