Some observations on compact indestructible spaces

Abstract

Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelöf T 2 indestructible space has the finite derived set property and a compact T 2 indestructible space is pseudoradial.A compact space is indestructible if it remains compact in any countably closed forcing extension. This is a particular case of the notion of Lindelöf indestructibility, whose study was initiated by Tall in [9]. A space is compact indestructible if and only if it is compact and Lindelöf indestrutible. A nice connection of Lindelöf indestructibility with certain infinite topological game was later discovered by Scheepers and Tall [8].G ω 1 1 (O, O) denotes the game of length ω 1 played on a topological space X by two players I and II in the following way: at the α-th inning player I choose an open cover U α of X and player II responds by taking an element U α ∈ U α . Player II wins if and only if {U α : α < ω 1 } covers X. Proposition 1. ( [8], Theorem 1) A Lindelöf space X is indestructibily Lindelöf if and only if player I does not have a winning strategy in G ω 1 1 (O, O).Recently, Dias and Tall [4] started to investigate the topological structure of compact indestructible spaces. In particular, they proved that a compact T 2 indestructible space contains a non-trivial convergent sequence ([4], Corollary 3.4).The aim of this short note is to strengthen the above result, by showing that indestructibility actually gives even more than sequential compactness (Theorem 3). However, indestructibility forces a compact space to be sequentially compact in the absolute general case, that is by assuming no separation axiom (Theorem 1). The same proof, with minor changes, will show that a Lindelöf T 2 indestructible space has the finite derived set property (Theorem 2).As usual, A ⊆ * B means |A \ B| < ℵ 0 (mod finite inclusion).1991 Mathematics Subject Classification. 54A25, 54D55, 90D44 .