1. Introduction
The born of the topology is remounted with the perspective realized by Gottfried Leibniz, who in the seventeenth century envisioned the geometria situs and analysis situs [1]. After one of the most famous problems called Leonhard Euler’s Seven Bridges of Königsberg problem and the polyhedron formula [2, 3] framework the formal initial study and the obtaining of the first results considering methods that after would give the graph theory [3], wherein a modern study associating algebra aspects is given the combinatorial topology, which establishes results-focused in the descomposibilities of vertices with certain invariance of co-dimensions on trees vertices as indirect graphs in which any two vertices are connected by exactly one path, thus using elements of commutative algebras with topology can be created results to its shell ability. Here also is more important the appearance of structures as topological groups to connect the vertex decomposability path complexes and the Dynkin graphs [4, 5, 6]. But the topology finds its greatest development when defined a topological space [7, 8] as a set endowed with a structure called topology which consists from a point of view purely orthodox of a system determined by metrics, norms, continuity elements, and defining continuous deformation of subspaces, where the deformations that are considered in topology are homeomorphisms [9, 10] and homotopies [11, 12]. Likewise, a property that is invariant under such deformations is a topological property. For example, the dimension concept, which endows a clear distinguishability between geometrical elements of different dimensions and the distinguishability between forms of two geometrical objects, can be made through compactness and connectedness. In a deep study arise the dimension theory [13, 14] as the topological theory of dimension of spaces.
However, the study of continuous applications and homeomorphisms legitimate the mappings between topological spaces and produces more specialized objects on the base of correspondences of these spaces, and under the resembling of Euclidean spaces near each point of a topological space called a manifold. Here, each point of an
Introducing algebra to study topological spaces, we obtain algebraic invariants that classify topological spaces up to homeomorphism, though usually are classified in a complete way with the homotopy equivalence [15].
2. From coverings, cobordisms, homotopies, and topology shepeers until the theories of Stone-Cech compactification, and others, with the theory of rings of continuous functions, and more
A covering is a local homeomorphism. This relation establishes different topological properties depending on the object or topological mapping constructed for establishing lifting, deck (covering) transformation group, regular covering [16],
Likewise, a universal covering [15]
The homotopy theory is a relative modern theory very useful in categories and schemes of categories, likewise as in the study of invariants as are homotopy groups, homology, and cohomology.
Today the most and biggest important problems in algebraic geometry and topology are focused on the study of higher categories, the
where here Brown’s representability theorem guarantees the existence of classifying spaces.
The idea of classifying space that classifies principal bundles can be generalized and inducted. In the case of generalized cohomology, the classification is realized on the base contravariant functors from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory.
Likewise, for example in derived categories problems, the classifying space can be determined through a spectrum
In a classic sense to the compactness [28] a Lindelöf space [28, 29] results in a suitable topological space due to that in a notion of compactness is required the existence of a finite sub-cover. Then in measure theory to some measure fields as the
From specialized studies of compactification to establish universal mappings between topological spaces and compact Hausdorff spaces, arise fundamental concepts to functionality and the universal property. Likewise, a compact Hausdorff space can be characterized through homeomorphisms and their topological measure invariants can be used in integration theory both in integrals over spaces of infinite dimension, as integration in chains (giving place to the integral topology) and integral transforms.
However, in the compactificación problem arise specialized studies of intervals and sets in fuzzy theory and other kinds of sets to neutrosophic sets [30, 31]. Likewise, considering
After, the separability is classified due to the separation axioms in topological spaces, having several classes of
Interesting embedding images of spaces arise called embedding separable metric spaces [28], for example, a separable metric space is homeomorphic to a subset of the Hilbert cube. In contraposition, also are of importance the non-separable spaces [29], where the elements of the space can be of bounded variation, or equipped with uncountable ordinals or even the spaces
3. Other specialized topology studies
Inside the differential topology, in the differentiable manifold context are developed many topological extensions of concepts as embedding, immersion, submersion, transversality, inclusion, epimorphism, diffeomorphisms, etc.
Likewise, the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions [33, 34] whose total space is a Kähler, nearly Kähler manifold, are studied through certain conditions that make totally geodesic mappings. This is important in the aspects of connections and umbilical points of the Kähler manifolds [33], likewise, also the investigated conditions for the semi-invariant submersion to be a Clairaut mapping.
Also, the geometric topology is fundamental to study in the ambit of the high-dimensional topology [35], the characteristic classes [36] as the basic invariant, and surgery theory, which conform a key theory, in aspects more subtly, accurate, and specialized of the general topology.
Some generalizations have been mentioned considering beyond of “set of points” which could not be available. Then we consider the concept of the lattice of open sets as the basic notion of the theory [37]. This notion is an alternative, for example of the Grothendieck topologies [38] which are structures defined on arbitrary categories that allow the definition of sheaves (germ spaces) on those categories, and such that give the definition of general cohomology theories [39] or even specialized themes of cohomology as motivic cohomology of tensor cohomologies.
The applications are diverse, though in computer science and physics find more application in the topological data analysis [40, 41] and topological quantum field theory [42, 43] where knot theory [41], the theory of four-manifolds in algebraic topology [44], and to the theory of moduli spaces [45] regain importance.
References
- 1.
Poincaré H. Analysis situs. Journal de l'École Polytechnique. 1895; 1 (2):1-123 - 2.
Aigner M, Ziegler GM. Three Applications of Euler’s Formula. Ch. 10 in Proofs from the Book. Berlin: Springer-Verlag; 1998 - 3.
Coxeter HSM. Euler’s Formula and “Poincaré’s Proof of Euler’s Formula.” §1.6 and Ch. 9 in Regular Polytopes. 3rd ed. New York: Dover; 1973. pp. 9-11 and 165-172 - 4.
Bondi A, Murray USR. “Graph Theory.” North-Holland. N. Y., USA; 1976 - 5.
Diestel R. Graph Theory. Graduate Texts in Mathematics Book Series. N. Y., USA: Springer; 2017 - 6.
Humphreys JE. 48. Fundamental domain § Affine reflection groups. In: Reflection Groups and Coxeter Groups. United Kingdom: Cambridge University Press; 1990 - 7.
Berge C. Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. New York: Dover; 1997 - 8.
Hausdorff F. Grundzüge der Mengenlehre. Leipzig, Germany: von Veit; 1914 - 9.
Dijkstra Jan J. On homeomorphism groups and the compact-open topology. The American Mathematical Monthly. 2005; 112 (10):910. DOI: 10.2307/30037630 - 10.
Willard S. General Topology [1970]. 1st ed. Mineola, N.Y: Dover Publications; 2004 - 11.
Aubry M. Homotopy Theory and Models. Birkhäuser: Boston, MA; 1995 - 12.
Collins GP. The shapes of space. Scientific American. 2004; 291 :94-103 - 13.
Eisenbud D. Commutative algebra. In: With a view toward algebraic geometry, Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag; 1995 - 14.
Weibel CA. An Introduction to Homological Algebra. Cambridge, United Kingdom: Cambridge University Press; 1995 - 15.
Hatcher A. Algebraic Topology. Cambridge: Cambridge University Press; 2002 - 16.
Brazas J. Semicoverings: A generalization of covering space theory. Homology, Homotopy and Applications. 2012; 14 (1):33-63. arXiv:1108.3021 - 17.
Taylor RL. Covering groups of nonconnected topological groups. Proceedings of the American Mathematical Society. 1954; 5 :753-768. DOI: 10.1090/S0002-9939-1954-0087028-0 - 18.
Brown R, Mucuk O. Covering groups of nonconnected topological groups revisited. Mathematical Proceedings of the Cambridge Philosophical Society. 1994; 115 (1):97-110. arXiv:math/0009021 - 19.
König W, Sprekels J. Karl Weierstraß (1815–1897): Aspekte seines Lebens und Werkes. Berlin, Germany: Springer-Verlag; 2015. pp. 200-201 - 20.
Mazza C, Voevodsky V, Weibel C. Lecture Notes on Motivic Cohomology. Vol. 2. Cambridge MA, USA: AMS, Clay Mathematics Institute; 2006 - 21.
Beke T. Sheafifiable homotopy model categories. Mathematical Proceedings of the Cambridge Philosophical Society. 2000; 129 (3):447-473. arXiv:math/0102087 - 22.
Dwyer WG, Spaliński J. Homotopy theories and model categories. In: Handbook of Algebraic Topology. Amsterdam: North-Holland; 1995. pp. 73-126 - 23.
Michael FA. Bordism and cobordism. Mathematical Proceedings of the Cambridge Philosophical Society. 1961; 57 :200-208 - 24.
Milnor J. A survey of cobordism theory. L’Enseignement Mathématique. 1962; 8 :16-23 - 25.
Brown R. Groupoids and crossed objects in algebraic topology. Homology, Homotopy and Applications. 1999; 1 :1-78 - 26.
Čech E. Höherdimensionale Homotopiegruppen. Zürich: Verhandlungen des Internationalen Mathematikerkongress; 1932 - 27.
Kamps KH, Porter T. Abstract homotopy and simple homotopy theory. River Edge, NJ: World Scientific Publishing; 1997. DOI: 10.1142/9789812831989 - 28.
Willard S. General Topology. New York, USA: Dover Publications; 1970. ISBN 0-486-43479-6 - 29.
Engelking R. General Topology. Berlin: Heldermann Verlag; 1989 - 30.
Blessie Rebecca S, Francina Shalini A. Neutrosophic generalized regular contra continuity in neutrosophic topological spaces. Research in Advent Technology. 2019; 7 (2):761-765. E-ISSN:2321-9637 - 31.
Dhavaseelan R, Jafari S. Generalized neutrosophic closed sets. New trends in Neutrosophic Theory And Applications. 2017; 2 :261-273 - 32.
Willard S. General topology. Reading, Mass: Addison-Wesley Pub. Co.; 1970. ISBN 0-486-43479-6 - 33.
Nomizu K. Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics. Vol. II. N.Y., USA: Interscience Publishers, Wiley & Sons; 1969 - 34.
Sahin B. Semi-invariant submersion from almost Hermitian manifolds. Canadian Mathematical Bulletin. 2011; 56 :173-182 - 35.
Sher RB, Daverman RJ. Handbook of Geometric Topology. North-Holland; 2002 - 36.
Chern S-S. Complex Manifolds Without Potential Theory. New Jersey, USA: Springer-Verlag Press; 1995 - 37.
Birkhoff G. Lattice Theory. Vol. 25. New York, NY: AMS Colloquium Publications; 1940. ISBN 978-0821810255 - 38.
Artin M. Grothendieck Topologies. Cambridge, MA: Harvard University, Dept. of Mathematics; 1962 - 39.
Switzer R. Algebraic Topology—Homology and Homotopy. New York, USA: Springer-Verlag; 1975. ISBN 3-540-42750-3 - 40.
Zadeh LA. “Fuzzy sets,” In: Information and control. Vol. 8. Cambridge, United Kingdom: Cambridge University Press; 1965. pp. 338–353 - 41.
Deo N. Graph Theory with Applications to Engineering and Computer Science. New Jersey, USA: Prentice-Hall Inc; 1974 - 42.
Gukov S, Kapustin A. Topological quantum field theory, nonlocal operators, and gapped phases of gauge theories. Scimago. Journal of High Energy Physics Theory. (United Kingdom: Scimago). 2013;arXiv:1307.4793 - 43.
Witten E. Topological quantum field theory. Communications in Mathematical Physics. 1988; 117 (3):353-386 - 44.
Maunder CRF. Algebraic Topology. London: Van Nostrand Reinhold; 1970. ISBN 0-486-69131-4 - 45.
Harris J, Morrison I. Moduli of curves. In: Graduate Texts in Mathematics. Vol. 187. New York: Springer Verlag; 1998
Notes
- Noetherian topological space.
- T0 (Kolmogorov); T1 (Fréchet); T2 (Hausdorff); T2½ (Urysohn) completely; T2 (completely Hausdorff); T3 (regular Hausdorff); T3½ (Tychonoff); T4 (normal Hausdorff); T5 (completely normal Hausdorff); T6 (perfectly normal Hausdorff).