It is well known the theoretical applications of generalized open sets in topological spaces, for example we can by them define various forms of continuous maps, compact spaces, separation axioms, etc. My question is: what the practical (reality) applications of generalized open sets such as semi-open and pre-open sets?.
Are these sets used to modeling some phenomena or problems?
In a loose language the semi open set is a set A which is neither closed nor open in in the standard terminology. You may think about it as of a plot of land fenced around the boundary with the flawed fence which has holes in it. So your plot of land is neither open nor closed. Regarding applications, the first what comes to my mind is the fundamental problem of magnetic hot plasma confinement in TOKOMAK torus like thermonuclear fusion reactor. If magnetic trap would be perfect ( no routes of escape) than the thermonuclear fusion would be sustainable. In other physical situations where there are present "passages to infinity" , at some conditions, we also face the problem of non perfect boundary, e.g. semi transparent mirror . You may think of your own examples. Regards, Georg von Brzeski
Thank you Prof. Georg von Brzeski
Please, If you have research or articles on this subject send it to me
As far as I know an abstract set theory is not what physicists love. The suggestions I shared with you are my own. You can do your own research. The main idea here ,in application to physics, as i see it, is as follows. You can assign the logic value 0 to an open set and the logic value 1 to a closed set. That may be something like a digital voltage levels. Then between voltage LOW (0) and voltage HIGH (1) there is a continuum of intermediate voltages ,which is a domain of analog signaling. It seems to be very interesting to me but just right now I am working on something else which I have to finish. I wish you every success in your research.
In addition to the excellent responses already given by @ J. G. von Brzeski , there is a bit more to add.
An application of pre-open sets in terms of chaos, solitons and fractals is given in
Article Some functions defined by α-open and preopen sets
A thorough look at a particular form of semi-open sets on topological spaces (with lots of examples) is given in
Article On supra R-open sets and some applications on topological spaces
More to the point, a good hunting ground for applications of semi-open sets is in the study of semi-open boundaries such as those found in
https://arxiv.org/pdf/astro-ph/0012137.pdf
Thank you Prof. James F Peters for your useful response.
I note that the generalized open sets are utilized to define new forms of continuous maps, compact spaces, separation axioms, etc., and by them we obtain various interesting theoretical results. But what then?. because this topics occupy a significant area of researches on topological domain, I will endeavor to find a practical application for them.
If there is no applications at the present time ( or we are unaware of such an applications) that does not mean that mathematical constructions are useless for applied science. I personally like what Polish science fiction writer Stanislaw Lem wrote about relation between pure mathematics and other sciences. Suppose there is a crazy Taylor who makes all kinds of apparel. With one button, without buttons, jackets with just one sleeve , without sleeves etc,etc. Taylor makes those apparel accordingly to his own rules and then he puts everything to a storage. A bunch of customers come to that storage and dig in the piles of that strange apparel and sometimes one can find what he is looking for. Now ,the Taylor is of course MATHEMATICS , customers are Physicists, Biologists, Engineers, Example: When matrix calculus was developed it was not immediately clear what will be benefit of it. Then Werner Heisenberg , digging in the Math repository found that matrix calculus was that tool he was looking for in developing his version of quantum mechanics (QM). Mathematicians are smart people. If we have problems to see how mathematics can benefit the world , that is due to our ignorance , I guess. Happy New 2018 Year for all creatures big and small.
Excellent answer Prof. J. G. von Brzeski
What we present today as a theoretical, may be applicable by others in future.
Who knows?
Excellent answer Prof. J. G. von Brzeski
What we present today as a theoretical, may be applicable by others in future.
Who knows?
I asked this question three years ago; at that time I was very excited to integrate theoretical concepts with real-life issues. To some extent, the answer is obvious now. In fact, the theoretical applications of generalizations of open sets are more than we count; it is enough they enable us to reformulate all concepts and examine the validity of their behaviors compared to their counterparts. Many times, we obtain unanticipated properties; for example, every topological space is T1 according to somewhere dense sets which is one of the generalizations of open sets; see, https://www.researchgate.net/publication/325597797_Somewhere_dense_sets_and_ST1-spaces.
Regarding the core inquire of the question, there are some valuable applications in published papers for these generalizations in the areas of digital topology (semi-open sets), analysis images (pre-open sets) as well as improvement of the approximations and accuracy measures of rough sets.
There are many applications of them in several fields such as (Rough Sets, Granular Computing and Computer Science). Also, many authors have been made applications of them in several fields such as Medical diagnosis. In Medical diagnosis, the doctors need some sorts to make an accurate decision about the infected people. For this line see the following papers:
https://www.researchgate.net/publication/350133840_Soft_ss-rough_sets_and_its_application_to_determine_COVID-19
https://www.researchgate.net/publication/344713860_Soft_pre-rough_sets_and_its_applications_in_decision_making
https://www.researchgate.net/publication/309814858_On_j-near_concepts_in_rough_sets_with_some_applications
Also, in the other fields such as
Largeron C, Bonnevay S (2002) A pretopological approach for structural analysis. Inf Sci 144:185–196
Stadler, B. M. R., Stadler, P. F., & Wagner, G. P. (2001). The topology of the possible: Formal spaces underlying patterns of evolutionary change. Journal of Theoretical Biology, 213, 241–274. http://dx.doi.org/10.1006/jtbi.2001.2423
The recent article: [T. M. Al-shami, Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets, Soft Computing, (2021), https://doi.org/10.1007/s00500-021-06358-0] presented practical applications of generalized open sets (somewhere dense sets) to improve the approximations and accuracy measures of a rough set.
Could you make a web page about those applications? I know a topologist that is happy to find applications of topology in functional analysis (so another branch of pure mathematics). Real world applications would be a revolution.
Dear Santanu Acharjee
I agree with your first two paragraphs, but what you wrote in the third paragraph is inaccurate; especially, the phrase " Molodtsov strictly wrote in several papers... and related ideas are also void."
I want to explain the following:
1- The shortcomings (or weaknesses) of Maji et al.'s concepts were corrected and improved in several succeeding articles (see, the recent one: Article T -soft equality relation
.2- The soft topology is not defined by using (only) Maji et al.'s concepts.
3- Which one of soft topologies do you mean? In the literature, there are two followed techniques to define soft topologies.
Also, I wonder what are the core concepts of soft set? As we know a soft set is a mapping without additional conditions.
Could you give examples (or any illustrations) to support your claims?
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