International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 15, Pages 2421-2427
doi:10.1155/IJMMS.2005.2421
Two classes of locally compact sober spaces
1Département des Mathématiques, École Supérieure des Sciences et Techniques de Tunis, 5 Avenue Taha Hussein, BP 56, Bab Mnara 1008, Tunisia
2Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire, El Manar II 2092, Tunisia
3Department of Mathematics, Institute of Multimedia, Route Mharza Km 1, 5, BP 1030, Sfax 3018, Tunisia
Received 15 December 2004; Revised 6 July 2005
Copyright © 2005 Karim Belaid et al. 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.
Abstract
We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.