i am trying to write a new open (may be which weak /strong)sets.may i know how many open sets (like alpha,beta,gamma, weak,pre etc) available in general topology. can anyone tell me list of open sets.
Your question is not clear enough. Do you ask about generalized topologies in the sense of Csaszar?
dear prof artur
i am asked for various types of open sets in general topology. not generalized topology.
Many kinds (like semiopen, preopen, alpha-, beta-, gamma-, ..., -open) of sets are considered in generalized topology in the sense of Csaszar. Hundreds of articles are already published in this topic, many of them in Acta Mathematica Hungarica.
You might find useful the following two survey articles regarding your Question:
(1) G. B. Navalag, Definition Bank"in General Topology, preprint, 2000, p. 1-50
Link: http://at.yorku.ca/i/d/e/b/75.htm
(2) David Gauld, Sina Greenwood and Ivan Reilly, On variations of continuity, Topology Atlas Survey, 1999, p.1-54.
link: http://at.yorku.ca/t/a/i/c/32.htm
There are also restrictions on the notion of "open set." In addition to open sets that are closed as well (clopen sets), there are the regular open sets: one of these is equal to the interior of its closure. Clopen sets are regular open, but not conversely (as the complement of a point in--say--the real line will attest). There are also the dense open sets, of importance in the study of Baire category issues. Obviously a dense open set is regular open iff it is not a proper subset. G-delta sets are also of interest in the Baire category context, as well as in analysis and descriptive set theory. They are a kind of "generalized open set," being countable unions of open sets. I'm not sure this is of any help, so I'll stop here.
The author needs different kinds of open open. if A is a collection of sets obtained by some rule and A is properly contained in the given topology(elements of A are open sets), then this is a collection of new kind of open sets. Best well known collection is the family of sets generated by the family of regular open sets. Find similar collection of sets.
- Badges
- Science topic
- Similar topics
- Geoscience
- Cartography
- Mapping
- Maps