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- W551460 abstract "A direction in a Type I space $X=cup_{alpha<omega_1}X_alpha$ is a closed and unbounded subset $D$ of $X$ such that given any continuous $f:Xtomathbb{L}_{ge 0}$ (the closed long ray), if $f$ is unbounded on $D$ then $f$ is unbounded on each unbounded subset of $D$. A closed copy of $omega_1$ is a direction in any Type I space. We study various aspects of directions and show some independence results. A sample: There is an $omega$-bounded Type I space without direction; PFA implies that a locally compact countably tight $omega_1$-compact Type I space contains a direction; if there is a Suslin tree then there is an $omega_1$-compact Type I manifold without direction; there are Type I first countable spaces which contain directions and whose closed unbounded subsets contain each a closed unbounded discrete subset. We also study a naturel order on the directions of a given space and show that we may obtain various classical ordered types with the space a manifold (often $omega$-bounded)." @default.
- W551460 created "2016-06-24" @default.
- W551460 creator A5059830643 @default.
- W551460 date "2014-04-04" @default.
- W551460 modified "2023-09-27" @default.
- W551460 title "Directions in Type I spaces" @default.
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