CONNECTIVE SPACES 1. Connective Spaces 1.1. Introduction. As
... is an equivalent formulation of a theorem of Kuratowski [6]. Finer topologies (with more open sets) induce finer connectologies (with less connected sets). A topology is an additional structure on top of a connectology: later it will be shown that homeomorphic topological spaces must have ‘isomorphi ...
... is an equivalent formulation of a theorem of Kuratowski [6]. Finer topologies (with more open sets) induce finer connectologies (with less connected sets). A topology is an additional structure on top of a connectology: later it will be shown that homeomorphic topological spaces must have ‘isomorphi ...
4. Topologies and Continuous Maps.
... Remark. Note that the concept of subbasis makes it now a trivial matter to construct topologies for a set X. For this we simply give ourselves any collection A of subsets of X whose union is X. This already defines a unique topology, given by first taking all intersections of elements of A and then ...
... Remark. Note that the concept of subbasis makes it now a trivial matter to construct topologies for a set X. For this we simply give ourselves any collection A of subsets of X whose union is X. This already defines a unique topology, given by first taking all intersections of elements of A and then ...
Point-Set Topology Definition 1.1. Let X be a set and T a subset of
... product topology on X ×Y is the smallest topology on X ×Y for which prX and prY are continuous. The functions prX and prY are called projections. Definition 2.10. Let A be a finite or countably infinite set and let (Xa , Xa ) be a topological space for each a ∈ A. Let Z = ×a∈A Xa . Let B = {×a∈A Ua ...
... product topology on X ×Y is the smallest topology on X ×Y for which prX and prY are continuous. The functions prX and prY are called projections. Definition 2.10. Let A be a finite or countably infinite set and let (Xa , Xa ) be a topological space for each a ∈ A. Let Z = ×a∈A Xa . Let B = {×a∈A Ua ...
Connectedness in fuzzy topology
... recognition, decision problems, function approximation, system theory, fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc. [2], [7]. In this note, our interests are in the study of certain concepts in fuzzy topology. The concepts of co ...
... recognition, decision problems, function approximation, system theory, fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc. [2], [7]. In this note, our interests are in the study of certain concepts in fuzzy topology. The concepts of co ...
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.