In my chinese book, it offers an example: there exist a separable metric space $X$ and a Lindelöf space $Y$, such that $X \times Y$ is not a Lindelöf space. However, the construction of this space is unclear, and I think maybe something is wrong. Does somebody know this example?
2026-04-30 09:05:02.1777539902
The product of a separable metric space and a Lindelöf space need not be Lindelöf
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The spaces $X\,'$ and $Y\,'$ in the next-to-last paragraph of this classic paper by E.A. Michael are respectively regular Lindelöf and metrizable, and their product is not Lindelöf. Dan Ma explains a similar example in this post to his Topology Blog.