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Abstract

By analyzing Dow’s construction, we introduce a general construction of regular Lindelöf spaces with points $G_\delta $. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindelöf P-space of pseudocharacter $\le \omega _1$ and of size $ \gt 2^\omega $, (2) CH and $\square (\omega _2)$ hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindelöf space with points $G_\delta $ and of size $ \gt 2^\omega $. This shows that, under CH, the non-existence of such a Lindelöf space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindelöf space with points $G_\delta $ and of size at least $(2^{\omega _1})^V$.