Sixteen new types of topologies have been introduced in the last years (2019-2024), such as: NonStandard Topology, Largest Extended NonStandard Real Topology, Neutrosophic Triplet Weak/Strong Topologies, Neutrosophic Extended Triplet Weak/Strong Topologies, Neutrosophic Duplet Topology, Neutrosophic Extended Duplet Topology, Neutrosophic MultiSet Topology, NonStandard Neutrosophic Topology, NeutroTopology, AntiTopology, Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, SuperHyperTopology, and Neutrosophic SuperHyperTopology: http://fs.unm.edu/TT/RevolutionaryTopologies.pdf

            1)  Refined Neutrosophic Topology

The neutrosophic set has been extended to the Refined Neutrosophic Set (Logic, Probability) [1], where there are multiple parts of the neutrosophic components, as such T was split into subcomponents T₁, T₂, ..., T_p, and I into I₁, I₂, ..., I_r, and F into F₁, F₂, ...,F_s, with p + r + s = n ≥ 2 and integers t, r, s ≥ 0 and at least one of them is ≥ 2.  Even more: the subcomponents T_j, I_k, and/or F_l can be countable or uncountable infinite subsets of [0, 1].

This definition also includes the Refined Fuzzy Set, when r = s = 0 and p ≥ 2; and the definition of the Refined Intuitionistic Fuzzy Set, when r = 0, and either p ≥ 2 and s ≥ 1, or p ≥ 1 and s ≥ 2.

All other fuzzy extension sets can be refined in a similar way.
The Refined Neutrosophic Topology is a topology defined on a Refined Neutrosophic Set.

{Similarly, the Refined Fuzzy Topology is defined on a Refined Fuzzy Set, while the Refined Intuitionistic Fuzzy Topology is defined on a Refined Intuitionistic Fuzzy Set.

And, as a generalization, on any type of fuzzy extension set [such as" Pythagorean Fuzzy Set, Spherical Fuzzy Set, Fermatean Fuzzy Set, etc.] one can define a corresponding fuzzy extension topology.}

          2) Refined Neutrosophic Crisp Topology

The Neutrosophic Crisp Set was defined by Salama and Smarandache in 2014 and 2015.

Let X be a non-empty fixed space. And let D be a Neutrosophic Crisp Set [2], where D = <A, B, C>, with A, B, C as subsets of X.

Depending on the intersections and unions between these three sets A, B, C one gets several:

Types of Neutrosophic Crisp Sets [2,3]

The object having the form D = is called:

(a) A neutrosophic crisp set of Type 1 (NCS-Type1) if it satisfies: A∩ B = B∩ C = C∩ A = (empty set).

(b) A neutrosophic crisp set of Type 2 (NCS-Type2) if it satisfies: A∩ B = B∩ C = C∩ A = and A∪ B ∪ C = X.

(c) A neutrosophic crisp set of Type 3 (NCS-Type3) if it satisfies: A∩ B ∩ C = and A∪ B ∪ C = X.

Of course, more types of Neutrosophic Crisp Sets may be defined by modifying the intersections and unions of the subsets A, B, and C.

    The Refined Neutrosophic Crisp Set was introduced by Smarandache in 2019, by refining/splitting D (and denoting it by RD = Refined D) by refining/splitting its sets A, B, C into sub-subsets as follows:

RD = (A₁, ..., A_p; B₁, ..., B_r; C₁, ..., C_s), with p, r, s ≥ 1 be positive integers and at least one of them be ≥ 2, and

and many Types of Refined Neutrosophic Crisp

          Therefore, the Refined Neutrosophic Crisp Topology is a topology defined on the Refined Neutrosophic Crisp Set.

 3) NeutroTopology [3] is a topology that has at least one topological axiom which is partially true, partially indeterminate, and partially false, 

or (T, I, F), where T = True, I = Indeterminacy, F = False, and no topological axiom is totally false, in other words:

 , where (1, 0, 0) represents the classical Topology, while (0, 0, 1) represents the below AntiTopology.

Therefore, the NeutroTopology is a topology in between the classical Topology and the AntiTopology.

4) AntiTopology [3] is a topology that has at least one topological axiom that is 100% false (T, I, F) = (0, 0, 1).

The NeutroTopology and AntiTopology are particular cases of NeutroAlgebra and AntiAlgebra [3] and, in general, they all are particular cases of the NeutroStructure and AntiStructure respectively, since we consider "Structure" in any field of knowledge [4].

5) SuperHyperTopology [5] is a topology build on the n^th-PowerSet of a given non-empty set H, that excludes the emptyset.

Therefore:

is the first powerset of the set H, without the emptyset (Ø);

is the second powerset of H (or the powerset of the powerset of H), without the emptysets;

and so on, the n-th powerset of H,

, where is repeated n times ( n ≥ 2 ), and without the emptysets.

6) Neutrosophic SuperHyperTopology [5] is, similarly, a topology build on the n^th-PowerSet of a given non-empty set H, but includes the emptysets [that represent indeterminaciesy] too.

As such, in the above formulas, that excludes the emptyset, is replaced by that includes the emptyset.

is the first powerset of the set H, including the emptyset (Ø);

is the second powerset of H (or the powerset of the powerset of H), that includes the emptysets;

and so on, the n-th powerset of H,

, where P is repeated n times ( n ≥ 2 ), and includes the emptysets

1-16) See all sixteen of them herein: http://fs.unm.edu/TT/RevolutionaryTopologies.pdf

Conclusion

These sixteen new types of topologies were introduced by Smarandache in 2019-2024, but they have not been much studied and applied yet, except the NeutroTopology and AntiTopology which got some attention from researchers.

References

[1] F. Smarandache, n-valued Refined Neutrosophic Set and Logic and its Applications in Physics, Progress in Physics, 143-146, Vol. 4, 2013, https://arxiv.org/ftp/arxiv/papers/1407/1407.1041.pdf and http://fs.unm.edu/RefinedNeutrosophicSet.pdf

[2] A.A. Salama, F. Smarandache, Neutrosophic Crisp Set Theory, Educational Publisher, Columbus, Ohio, USA, 2015; http://fs.unm.edu/NeutrosophicCrispSetTheory.pdf 

[3] Florentin Smarandache, Refined Neutrosophic Crisp Set (RNCS), in the book Nidus Idearum, pp. 114-116, Vol. VII, third edition, 2019, Editions Pons, Brussels, Belgium;

Nidus Idearum book:

http://fs.unm.edu/NidusIdearum7-ed3.pdf

Refined Neutrosophic Crisp Set (RNCS) – chapter:

http://fs.unm.edu/RefinedNeutrosophicCrispSet.pdf

[3] F. Smarandache, NeutroAlgebra & AntiAlgebra are generalizations of classical Algebras, http://fs.unm.edu/NA/NeutroAlgebra.htm and http://fs.unm.edu/NeutroAlgebra-general.pdf, 2019-2022.

[4] Florentin Smarandache, Structure, NeutroStructure, and AntiStructure in Science, International Journal of Neutrosophic Science (IJNS), Volume 13, Issue 1, PP: 28-33, 2020;  http://fs.unm.edu/NeutroStructure.pdf

[5] F. Smarandache, The SuperHyperFunction and the Neutrosophic SuperHyperFunction, Neutrosophic Sets and Systems, Vol. 49, 2022, pp. 594-600, http://fs.unm.edu/NSS/SuperHyperFunction37.pdf

[6] F. Smarandache, New Types of Topologies and Neutrosophic Topologies, Neutrosophic Systems with Applications, pp. 1-3, Vol. 1, 2023, http://fs.unm.edu/TT/NewTypesTopologies.pdf

[7] Florentin Smarandache, New Types of Topologies and Neutrosophic Topologies (Improved Version), Neutrosophic Sets and Systems, Vol. 57, 2023, pp. 234-244. DOI: 10.5281/zenodo.8271368

[8] Mohammed A. Al Shumrani, Florentin Smarandache: Introduction to Non-Standard Neutrosophic Topology. Symmetry 2019, 11, 0.

[9] Florentin Smaradnache, Foundation of Revolutionary Topologies: An Overview, Examples, Trend Analysis, Research Issues, Challenges, and Future Directions, Neutrosophic Systems with Applications, 45-66, Vol. 13, 2024.

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