A consistent example of a hereditarily $ \mathfrak{c}$-Lindelöf first countable space of size $ >\mathfrak{c}$

István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

Answering a question raised by Anishkievic and Arhangelski{\u{\i\/}}\kern.15em, we show that if $ V\vDash CH$ then there is an $ \omega_1$-closed and $ \omega_2-CC$ partial order $ P$ such that, in $ V^P$, there exists a 0-dimensional, $ T_2$, hereditarily $ \mathfrak{c}$-Lindelöf , and first countable space of cardinality $ \omega_2=\mathfrak{c}^+$. The question if there is such a space (even with ``hereditarily'' dropped) in ZFC remains open.

2000 Mathematics Subject Classification: 54A25, 54A35, 03E35

Key words and phrases: $ \mathfrak{c}$-Lindelöf, first countable, consistent example

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