Mizokami and Linʹs conjecture on σ- pseudo-bases

Mizokami and Lin conjectured in [T. Mizokami, S. Lin, On spaces with a σ- CF ∗ pseudo-base, Math. Jpn. 46 (1997) 377–384]: Every space X with a σ- CF ∗ pseudo-base satisfies that either X is an ℵ 0 -space or each compact subset of X is finite. In this paper we show this conjecture is true. Also we answer some other questions in the book “Open Problems in Topology, II”.