We give easy proofs that (a) the Continuum Hypothesis implies that if the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is a $D$-space, and (b) Borel's Conjecture implies every Rothberger space is Hurewicz.

Keywords:

productively Lindelöf, $D$-space, projectively $\sigma$-compact, Menger, Hurewicz productively Lindelöf, $D$-space, projectively $\sigma$-compact, Menger, Hurewicz

MSC Classifications:

54D20, 54B10, 54D55, 54A20, 03F50 show english descriptions Noncompact covering properties (paracompact, Lindelof, etc.)
Product spaces
Sequential spaces
Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
Metamathematics of constructive systems 54D20 - Noncompact covering properties (paracompact, Lindelof, etc.)
54B10 - Product spaces
54D55 - Sequential spaces
54A20 - Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
03F50 - Metamathematics of constructive systems

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