Character and tightness of hyperspaces with the Fell topology

✍ Scribed by Ji-Cheng Hou

• Book ID: 104295264
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 492 KB
• Volume: 84
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00092-8

✦ Synopsis

For a Hausdorff space X, we denote by 2x the collection of all closed subsets of X. The Fell topology 7.~ on 2x has as a subbase all sets of the form V-= {F E 2x: F n V # 8}, where L' is an open subset of X and of the form (Kc)+ = {F E 2x: F n K = 8}, where K is a compact subset of X. In this paper we prove that max{d,(X),kl,(X),t(X)} < t((2X,7~)) < max{&(X), k%(X), x(X)} and ~((2~ , m)) = max{d,(X), k/c,(X), x(X)}, where t(X) and x(X) denote the tightness and character of X, respectively, d,(X) is the smallest cardinal number T such that the density of every closed subset of X is less than or equal to 7: and k&(X) and k-k,(X) are two cardinal invariants related to the family of all compact subsets of X. We also obtain that for a locally compact space X the tightness of (2x, T-F) and the character of (2x, TF) coincide. We construct a space Y such that t((aY ,TF)) # max{d,(Y), k&(Y), x(Y)}. We also give an answer to a question of Beer (1993).