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submetrizable; absolutely submetrizable
We introduce a notion of absolute submetrizability (= ``every Tychonoff subtopology is submetrizable'') and investigate its behavior under basic topological operations. The main result is an example of an absolutely submetrizable space that contains an uncountable set of isolated points (hence the space is neither separable nor hereditarily Lindelöf). This example is used to show that absolute submetrizability is not preserved by some topological operations, in particular, by free sums.
[ENG] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, revised ed., 1989. MR 1039321 | Zbl 0684.54001
[MRO] Mrówka S.: On completely regular spaces. Fund. Math. 41 (1954), 105-106. MR 0063650
[SNE] Sneider V.: Continuous images of Souslin and Borel sets; Metrization theorems. Doklady Akad. Nauk SSSR 50 (1959), 77-79. MR 0053493
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