On the $$C_k$$Ck-stable closure of the class of (separable) metrizable spaces

By T. Banakh, Saak Gabriyelyan. 2014

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Denote by $$\mathbf {C}_k[\mathfrak M]$$Ck[M] the $$C_k$$Ck-stable closure of the class $$\mathfrak M$$M of all metrizable spaces, i.e., $$\mathbf {C}_k[\mathfrak M]$$Ck[M] is the smallest class of topological spaces that contains $$\mathfrak M$$M and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $$C_k(X,Y)$$Ck(X,Y) with Lindelöf domain in this class. We show that the class $$\mathbf {C}_k[\mathfrak M]$$Ck[M] coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces $$C_k(X,Y)$$Ck(X,Y) with a separable metrizable space X and a metrizable space Y. We say that a topological space Z is Ascoli if every compact subset of $$C_k(Z)$$Ck(Z) is evenly continuous; by the Ascoli Theorem, each k-space is Ascoli. We prove that the class $$\mathbf {C}_k[\mathfrak M]$$Ck[M] properly contains the class of all Ascoli $$\aleph _0$$ℵ0-spaces and is properly contained in the class of $$\mathfrak {P}$$P-spaces, recently introduced by Gabriyelyan and Kąkol. Consequently, an Ascoli space Z embeds into the function space $$C_k(X,Y)$$Ck(X,Y) for suitable separable metrizable spaces X and Y if and only if Z is an $$\aleph _0$$ℵ0-space.

Subfields: Geometry and Topology, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1295e81aaf2ffe3b0d33ec7c8fa2a1ea2b64c6f