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International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 7, Pages 387-399
http://dx.doi.org/10.1155/S0161171202202203

Neighborhood spaces

1Department of Mathematics, Washington State University, Pullman 99164-3113, WA, USA
2Department of Mathematics, Kangwon National University, Chuncheon 200-701, Korea

Received 7 February 2002

Copyright © 2002 Hindawi Publishing Corporation. 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

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more general p-stacks.