Algebraic Topology

... interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory. Preceding the four main chapters there is a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in ...

... interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory. Preceding the four main chapters there is a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in ...

FUZZY PSEUDOTOPOLOGICAL HYPERGROUPOIDS 1

... As the hyperoperation ”·” is a mapping from H × H to P ∗ (H) giving topologies on H and P ∗ (H) we can speak about the continuity of ”·”. Unfortunately, if a topology is given on H, there is no standard straightforward way of generating a topology on P ∗ (H). This leads to the following definition. ...

... As the hyperoperation ”·” is a mapping from H × H to P ∗ (H) giving topologies on H and P ∗ (H) we can speak about the continuity of ”·”. Unfortunately, if a topology is given on H, there is no standard straightforward way of generating a topology on P ∗ (H). This leads to the following definition. ...

Title of Paper (14 pt Bold, Times, Title case)

... fuzzy topological space and at the same time has the potential of generalizing the theory further. Also, due to the relaxation of the stringent condition regarding preservation of arbitrary supremum in this generalized structure, the results obtained by generalizing the existing theories may be appl ...

... fuzzy topological space and at the same time has the potential of generalizing the theory further. Also, due to the relaxation of the stringent condition regarding preservation of arbitrary supremum in this generalized structure, the results obtained by generalizing the existing theories may be appl ...

DECOMPOSITION OF CONTINUITY AND COMPLETE CONTINUITY

... In this section, fuzzy α-continuous and weak fuzzy AB-continuous functions are introduced. Decomposition of fuzzy continuous and complete fuzzy continuous functions are studied. Definition 3.1. Let (X, T ) and (Y, S) be any two smooth fuzzy topological spaces. A function f : (X, T ) → (Y, S) is said ...

... In this section, fuzzy α-continuous and weak fuzzy AB-continuous functions are introduced. Decomposition of fuzzy continuous and complete fuzzy continuous functions are studied. Definition 3.1. Let (X, T ) and (Y, S) be any two smooth fuzzy topological spaces. A function f : (X, T ) → (Y, S) is said ...

# Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a ""clockwise"" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a ""clockwise"" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as 20px in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image 20px.The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.