A regular Lindelöf semimetric space which has no countable network

Abstract:

A completely regular semimetric space $M$ is constructed which has no $\sigma$-discrete network. The space $M$ constructed has the property that every subset of $M$ of cardinality ${2^{{\aleph _0}}}$ contains a limit point of itself; thus, assuming ${2^{{\aleph _0}}} = {\aleph _1},M$ is Lindelöf. It is also shown from the same space $M$ that, assuming ${2^{{\aleph _0}}} = {\aleph _1}$, there exists a regular Lindelöf semimetric space $X$ such that $X \times X$ is not normal (hence not Lindelöf).