Topological Algebra
... (3) The sets GL(n, R), GL(n, C), and GL(n, H) of invertible n × n matrices with real, complex, and quaternionic entries, respectively, under the matrix multiplication. (4) SL(n, R), SL(n, C), O(n), O(n, C), U (n), SO(n), SO(n, C), SU (n), and other subgroups of GL(n, K) with K = R, C, or H. 26.7x. I ...
... (3) The sets GL(n, R), GL(n, C), and GL(n, H) of invertible n × n matrices with real, complex, and quaternionic entries, respectively, under the matrix multiplication. (4) SL(n, R), SL(n, C), O(n), O(n, C), U (n), SO(n), SO(n, C), SU (n), and other subgroups of GL(n, K) with K = R, C, or H. 26.7x. I ...
LOCAL MONODROMY OF BRANCHED COVERS AND DIMENSION
... As an application of Theorem 1, we also obtain a positive result in a special case of the conjecture of Church and Hemmingsen for branched covers having local multiplicity at most three. More precisely we have the following result. Theorem 3. Let f : M → N be a proper branched cover between n-manifo ...
... As an application of Theorem 1, we also obtain a positive result in a special case of the conjecture of Church and Hemmingsen for branched covers having local multiplicity at most three. More precisely we have the following result. Theorem 3. Let f : M → N be a proper branched cover between n-manifo ...
Modal logics based on the derivative operation in topological spaces
... For A ⊆ X , a point x ∈ X is a co-limit point of A iff there exists an open neighborhood B of x such that B ⊆ A ∪ {x} For A ⊆ X , t(A) is the set of co-limit points of A Remind that In(A) = A ∩ t(A) For A ⊆ X , a point x ∈ X is a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) ...
... For A ⊆ X , a point x ∈ X is a co-limit point of A iff there exists an open neighborhood B of x such that B ⊆ A ∪ {x} For A ⊆ X , t(A) is the set of co-limit points of A Remind that In(A) = A ∩ t(A) For A ⊆ X , a point x ∈ X is a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) ...
Normal induced fuzzy topological spaces
... following theorem. Theorem 2.12. The finite infima of NLSC functions is NLSC. Proof. Let ψ(x) = inf {φi (x)} , i = 1, 2, ..., n and x ∈ X, where each φi (x) is NLSC. We prove that ψ(x) is NLSC. Let for x ∈ X, ψ(x) < λ and U be an open set containing x. Now, ψ(x) < λ i.e. inf(φi (x)) < λ implies that ...
... following theorem. Theorem 2.12. The finite infima of NLSC functions is NLSC. Proof. Let ψ(x) = inf {φi (x)} , i = 1, 2, ..., n and x ∈ X, where each φi (x) is NLSC. We prove that ψ(x) is NLSC. Let for x ∈ X, ψ(x) < λ and U be an open set containing x. Now, ψ(x) < λ i.e. inf(φi (x)) < λ implies that ...
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.