Search results for "First-countable space"
showing 3 items of 3 documents
Cardinal estimates involving the weak Lindelöf game
2021
AbstractWe show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a wi…
Countable connected spaces and bunches of arcs in R3
2006
Abstract We investigate the images (also called quotients) of countable connected bunches of arcs in R 3 , obtained by shrinking the arcs to points (see Section 2 for definitions of new terms). First, we give an intrinsic description of such images among T 1 -spaces: they are precisely countable and weakly first countable spaces. Moreover, an image is first countable if and only if it can be represented as a quotient of another bunch with its projection hereditarily quotient (Theorem 2.7). Applying this result we see, for instance, that two classical countable connected T 2 -spaces—the Bing space [R.H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953) 474], and th…
A note on monolithic scattered compacta
2015
Abstract For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP, for short) is that the dual unit ball B E ⁎ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X turns out to be sufficient for C 0 ( X ) to enjoy the CSCP.