Large regular Lindelöf spaces with points $G_\delta $

Toshimichi Usuba¹

¹ Faculty of Science and Engineering Waseda University 3-4-1 Okubo, Shinjuku Tokyo 169-8555, Japan

Fundamenta Mathematicae, Tome 237 (2017) no. 3, pp. 249-260

Résumé

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 $(2^{\omega _1})^V$.

Détail

DOI : 10.4064/fm296-8-2016

Keywords: analyzing dow construction introduce general construction regular lindel spaces points delta using construction prove following suppose either there exists regular lindel p space pseudocharacter omega size omega square omega holds there exists kurepa tree there exists regular lindel space points delta size omega shows under non existence lindel space has large cardinal strength prove every forcing adding real creates regular lindel space points delta size least omega