International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 208693, 11 pages
Research Article

On Open-Open Games of Uncountable Length

Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland

Received 31 March 2012; Revised 8 June 2012; Accepted 8 June 2012

Academic Editor: Irena Lasiecka

Copyright © 2012 Andrzej Kucharski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this paper is to investigate the open-open game of uncountable length. We introduce a cardinal number 𝜇 ( 𝑋 ) , which says how long the Player I has to play to ensure a victory. It is proved that 𝑐 ( 𝑋 ) 𝜇 ( 𝑋 ) 𝑐 ( 𝑋 ) + . We also introduce the class 𝒞 𝜅 of topological spaces that can be represented as the inverse limit of 𝜅 -complete system { 𝑋 𝜎 , 𝜋 𝜎 𝜌 , Σ } with 𝑤 ( 𝑋 𝜎 ) 𝜅 and skeletal bonding maps. It is shown that product of spaces which belong to 𝒞 𝜅 also belongs to this class and 𝜇 ( 𝑋 ) 𝜅 whenever 𝑋 𝒞 𝜅 .