Strongly Screenableness and its Tychonoff Products

Authors: Jianjun Wang, Peiyong Zhu

Abstract:

In this paper, we prove that if X is regular strongly screenable DC-like (C-scattered), then X ×Y is strongly screenable for every strongly screenable space Y . We also show that the product i∈ω Yi is strongly screenable if every Yi is a regular strongly screenable DC-like space. Finally, we present that the strongly screenableness are poorly behaved with its Tychonoff products.

Keywords: Topological game, strongly screenable, scattered, Cscattered.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335290

References:

[1] J. Greever, On screenable topological spaces. Proc. Japan Acad. 44 (1968) 434-438.
[2] R. Telgarsky, C-scattered and paracompact spaces. Fund. Math. 88 (1971) 59-74.
[3] R. Telgarsky, Spaces defined by topological games. Fund. Math. 88 (1975) 193-223.
[4] Y. Yajima, Topological games and products III. Fund. Math. 117 (1983) 223-238.
[5] Y. Yasui, Generalized paracompactness. K. Morita, J. Nagata, Eds., Topics in General Topology. 1989, 161-202.
[6] T. Przymusinski, Normality and paracompactness in finite and countable Cartesian products. Fund. Math. 105 (1980) 87-104.
[7] F. Galvin and R. Telgarsky, Stationary strategies in topological games. Topology Appl. 22 (1986) 51-69.
[8] Y. Yajima, On the submetacompactness of products. Proc. Amer. Math. Soc. 107 (1989) 503-509.
[9] R. Engleking, Hereditarily screenableness and its Tychonoff products. General topology (Heldermann, Berlin, 1989).
[10] G. Gruenhage and Y. Yajima, A filter property of submetacompactness and its application to products. Topology Appl. 36 (1990) 43-55.
[11] H. Tanaka, Submetacompactness and weak submetacompactness in countable products. Topology Appl. 67 (1995) 29-41.
[12] H. Tanaka, Covering properties in countable products. Tsukuba J. Math. 2 (1993) 565-568.
[13] Z. Peiyong, Hereditarily screenableness and its Tychonoff products. Topology Appl. 83 (1998), 231-238.
[14] Z. Peiyong, Inverse limits and infinite products of expandable. Scientiae Mathematicae Japonicae. 65 (2007) 173-178.