-ith Murad Hussain & Moiz Uddin Khan & A Zemci Ozcelik, “Extending closed properties on Generalized Topological Groups ” Arabian Journal of Mathematics: Volume 3, Issue 3 (2014), Page 341-347 Doi: 10.1007/s40065-014-0102-9 Springer Berlin Heidelberg June 2014.

- With Moiz Uddin Khan and Murad Husain” On generalized topological groups”

FILOMAT 27: 4 (2013)  567-575.

- Cenap OZEL, Moiz Uddin Khan and Murad Husain” More Results On generalized topological groups” Creative Math Information 22(2013) No:1 47-51.

I think that, while working on a kind of a generalized topology, we will have to make it more precise, more clear to others in what sense a gts is generalized, whose approach to generalizations of topological spaces is taken into consideration in our current work.  Most mathematicians working on relatively simple Csaszar' s generalizanion of the notion of a topology do not seem to work on Delfs and Knebusch  more original idea of admissible open coverings. 

Dear Prof. Peters:

I am afraid your examples are mainly not new generalizations of the notion of a topological space (do you understand the definition of a topological space?).

Majority of the notions you gave belong to the traditional topology. So still I count two generalizations of the notion of a topology on a set. Maybe point-free topology could be considered as another generalization.

I think many generalization can found in literature. Another point ist they are independent to each other. I have used one generalization defined by A. Csaszar to construct generalized topological groups.

How many kinds of generalized topology were defined?. Available from: https://www.researchgate.net/post/How_many_kinds_of_generalized_topology_were_defined#view=552f8260d685cc2f058b456c [accessed Apr 16, 2015].

So am I right to say you are working in generalized topology in the sense of A. Csaszar? (Your definition of a generalized topological group is in the framework of the Csaszar's definition of a generalized topological space).

There are other notions like pretopological spaces (or pseudotopological spaces?), but how to classify them??

For example: I think that bitopological spaces should not be counted as a generalization, since it is the usual notion of a  topology taken twice.

How to count distinctive generalizations of usual topology?

The necessary (but not sufficient)  condition for a generalization should be the following: generalized topological spaces and their generalized continuous mappings form a category (GTS) and Top is a subcategory of this larger category.

There are plenty of notions of new types of open sets and new types of continuity in the usual topology, as in the topic

https://www.researchgate.net/post/how_many_open_sets_are_created_already_in_general_topology

They should not be counted as generalizations, since they can be produced from usual topological spaces.

Csaszar's generalization (= to avoid intersection axiom) of 2002 is already available in the 1950 book of A. Appert, K. Fan, Espaces topologiques intermediaires.

Generalizations of topologies include closure algebras i.e. Boolean algebras with closure operators [J.C.C. McKinsey & A. Traski (1944), R. Sikorski (1949)]. Related books: G. Nobeling, Grundlagen der Analytischen Topologie (1954); H.Rasiowa & R. Sikorski, The Mathematics of Metamathematics (1968).

Important generalizations of topologies include noncommutative topologies in which the power set has an additional binary not necessarily commutative operation which replaces the set-theoretic intersection [cf. F. Borceux & G. Van Den Bosche, An essay on non-commutative topology, Top. Appl. 31 (1989)].

This sort of generalizations can best be made in the framework of many-valued topology (= fuzzy topology = L-valued topology) where the complete lattice L is also a semigroup with extra properties when necessary; see the book by U. Hohle, Many Valued Topology and its Applications (2001). One important feature of this approach is that some non-topological events (e.g. non-topological convergences) become topological in this generalized setting.

And what about lax algebras or (T,V)-spaces? You need a quantale V and a monad T, and produce the category (T,V)-Cat. Now, chosing beta and 2 (as something like parameters), we get a category (beta,2)-Cat  isomorphic to Top.

I have learned about them recently from the book

"Monoidal topology" by Hoffmann, Seal and Tholen (CUP, 2014).

So, strictly speaking, we do not get a generalization but a parametrization?

Question: Are  generalized topological spaces in the sense of Delfs and Knebusch examples of lax algebras?

There exist also supertopological spaces, supernearness spaces, superneighbourhood spaces.(Gotchev, Doitchinov)

I am afraid that the spaces exist only in one's imagination.  There exist some descriptions of such spaces.

We can define as many as generalised topilogy as many weaker system of topological axioms can be find.

I am afraid that what mathematicians choose as axioms need not be true. If one deduces  from a sentence which is not necessarily true that an object X exists, one should not be sure that the object exists. It would exist if the assumptions under which its existence can be proved were satisfied, However, it is only a hypothesis that the assumptions are satisfied. One is allowed to assume that they are not satisfied.

Please see my paper for partial answer:

Generalized topologies on finite sets

Authors Mohammad Reza Ahmadi Zand Publication date 2016 Honam Math. J. Volume 38 Issue 3 Pages 455-465 Publisher 호남수학회 Description The number of topologies on a finite set is a famous open problem. In the present paper we discuss a method of obtaining the number of generalized topologies on finite sets. Total citations Cited by 2 2016

Now smopologies are another kind od generalized topologies.

Look into the project about Delfs-Knebusch generalized topology.

Another famous kind of generalized topology is called supra topology which introduced by Mashhore et al. in 1983 [A. S. Mashhour, A. A. Allam, F. S. Mahmoud and F. H. Kheder, On supra topological spaces, Indian Journal of Pure and Applied Mathematics, 14 (4) (1983) 502-510].

This type was defined by omitting the finite intersection condition of topology.

Later on, many mathematicians worked on the supra topologies and considered as a suitable frame for some real life applications.

Herein, you can see the original paper of supra topology.

The classical types of generalized topology have been defined by dropping one or more of the topology conditions. Another path to generalize topology has been investigated by adding a partial order relation to a topological structure. It is called a topological ordered space [L. Nachbin, Topology and ordered, D. Van Nostrand Inc. Princeton, New Jersey, (1965)]. By considering the crisp set is a special case of fuzzy sets and soft sets, we deal with fuzzy topology and soft topology as another generalization of topology.

Doesn't count, since everyday there is a new structure generalized topology is created such that fuzzy topology, multi topology, soft topology, supra topology, weak structures, minimal structures, ordered topology, fuzzy rough topology, multi fuzzy topology, soft multi topology, ...etc.

Regards.