Introductory Chapter: The Topology from Classic Studies until Its Last Frontiers

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 n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

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], G-covering [16, 17, 18] or even certain topological actions as the monodromy action [19], homomorphisms of fundamental groups, groupoids relations even Galois connection.

Likewise, a universal covering [15] $p:\mathrm{D}\to \mathrm{X},\text{is}$ regular, with the deck transformation group being isomorphic to the fundamental group ${\pi }_1\left(X\right).$

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 p-adic groups theory, symplectic geometries, spectrum and generalized cohomology, motivic cohomology on tensor derived categories, among others. Even in the representation theory to classifying spaces of vector bundles, the homotopy operations result in determinant, where for a topological group G, the classifying space for principal G-bundles s a space BG, such that, for each topological space X, we have

$$\left[X,\mathit{BG}\right]=\left\{\text{principal}G-\text{bundle}\mathrm{on}X\right\}/\sim ,\left[f\right]\mapsto f^{*}\mathit{EG},$$E1

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 ${\mathrm{Mod}}_B{\text{CRing}}_{\mathrm{A}//\mathrm{B}}$, which is defined by $X\mapsto {\Omega }_{X/\mathrm{A}}{\otimes }_{X}B.$ In particular, the image of $\mathrm{A}\to B\to \mathrm{B},$ under this functor is $B\mapsto {\Omega }_{X/\mathrm{A}}.$ As the important fact is necessary to consider that the derived tensor product is a regular tensor product [20]. Then the adjunction of categories $\mathrm{Ch}{\left(B\right)}_B\leftrightarrow s{\text{CRing}}_{A//B},$induces an adjunction to the level of homotopy categories [21, 22]. In this sense results very interesting the sphere homotopies used in generalized cohomology where the image of the functor is a sphere spectrum given, for example, by $S^0\to S^1\to S^2.$ In this study arise important results of the homotopy theory as the homotopy excision theorem, the suspension theorem, etcetera. In the sphere, homotopy result interesting the study of invariants obtained correlating the stable homotopy theory and cobordisms [23, 24] obtained through these invariants between spheres and their compact groups, or a class of compact manifolds of the same dimension, where cobordism is an equivalence relation on those compact manifolds. A concrete application of it, is the compute of cohomologies or homologies considering oriented cobordism and complex cobordism. Likewise, in algebraic topology, the cobordism is a calculation method to determine cohomologies, for example, the homotopy groups [25, 26, 27] of spheres where spheres of various dimensions can wrap around each other, and finally result are related by homotopy invariants and topological invariants of the homotopy groups. Likewise, homotopy groups such as ${\pi }_{n+k}\left(S^n\right),$result independent of the dimension sphere $\mathrm{n},$ for $n\ge k+2.$ These are the stable homotopy groups of spheres that form a coefficient ring of a special theory of cohomology called stable cohomology theory. The unstable homotopy groups $n<\mathrm{k}+2$no form a ring, however, these have been tabulated to $n<20.$ Modern calculations using spectral sequences (method whose fundamental is homotopy) obtain more patterns to tabulation of homotopy groups to calculations in cohomology.

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 $\sigma$-field results Lindelöf. However, topologically what is beyond of the compactness for example, strong compactness¹, para-compactness, or pre-compactness. Even in many cases the compactness requires its generalization.

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 $\mathbb{P}$, a non-empty fixed set, we can say that a neutrosophic set Ҥ on the universe $\mathbb{P}$ is defined as the space Ҥ$\left\{⟨p,\left(t_{Ҥ}\left(p\right),i_{Ҥ}\left(p\right),f_{Ҥ}\left(p\right)\right)⟩:\mathit{pЄP}\right\}$ where $t_{Ҥ}\left(p\right),i_{Ҥ}\left(p\right),f_{Ҥ}\left(p\right)$ represent the degree of membership function $t_{Ҥ}\left(p\right)$, the degree of indeterminacy $i_{Ҥ}\left(p\right)$ and the degree of non-membership function $f_{Ҥ}\left(p\right)$ respectively for each element $\mathit{pЄP}$ to the set Ҥ. Also, $t_{Ҥ},i_{Ҥ},f_{Ҥ}$: $\mathbb{P}\to {}^{-0,1^{+}}$ and $-$0 $\le t_{Ҥ}\left(p\right)+i_{Ҥ}\left(p\right)+f_{Ҥ}\left(p\right)\le 3^{+}$. The set of all Neutrosophic sets over $\mathbb{P}$ is denoted by N_eu($\mathbb{P}$). Then is a generalization of the classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, and paraconsistent set.

After, the separability is classified due to the separation axioms in topological spaces, having several classes of $T_i$-separability with $i=0,1,2,21/2,3,31/2,4,5,6$². Then different topological spaces are generated on the base of the separation axioms [32], and some other property related with every pair of topologically distinguishable points [28], for example, an accessible space or a space with Fréchet topology. Likewise, in any topological space we have, as properties of any two distinguishable points, the implications:

$$\text{Separated}\Rightarrow \text{topological distinguishable}\Rightarrow \text{distinct topological space},$$E2

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 $L^{\infty }.$Exist more non-separable spaces which relate metric spaces of density equal to infinite cardinals α, which are isometric to subspaces of continuous functions as $C\left({\left[0,1\right]}^{\alpha },R\right).$

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.