Affine Decomposition of Isometries in Nilpotent Lie Groups
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
A study on compactness in metric spaces and topological spaces
... Let f: X→Y be continuous, where Y is an ordered set in the order topology. If x is compact, then there exit points c and d in X such that f(c)≤f(x)≤f(d) for every x∈X. The extreme value theorem of calculus is the special case of this theorem that occurs when we take X to be a closed interval in ℝ an ...
... Let f: X→Y be continuous, where Y is an ordered set in the order topology. If x is compact, then there exit points c and d in X such that f(c)≤f(x)≤f(d) for every x∈X. The extreme value theorem of calculus is the special case of this theorem that occurs when we take X to be a closed interval in ℝ an ...
ON WEAKLY e-CONTINUOUS FUNCTIONS
... A subset A of a space X is called regular open [23] (resp. regular closed [23]) if A = int(cl(A)) (resp. A = cl(int(A))). A subset A of a space X is called δ-semiopen [19] (resp. preopen [12], δ-preopen [22], α-open [14], semi-preopen [3] or β-open [1], b-open [2] or sp-open [5] or γ-open [8], e-ope ...
... A subset A of a space X is called regular open [23] (resp. regular closed [23]) if A = int(cl(A)) (resp. A = cl(int(A))). A subset A of a space X is called δ-semiopen [19] (resp. preopen [12], δ-preopen [22], α-open [14], semi-preopen [3] or β-open [1], b-open [2] or sp-open [5] or γ-open [8], e-ope ...
![arXiv:math/0201251v1 [math.DS] 25 Jan 2002](http://s1.studyres.com/store/data/014913155_1-788f445ae3a0c155aa768d6df676e28b-300x300.png)
Sequential Separability vs Selective Sequential
... n ∈ ω such that the set T = Sn ∩ [A]seq is infinite. Since T is a sequence converging to ∞ in X, we finally have [[A]seq ]seq = clX (A). So, X is a sequential space of sequential order 2. Fact 2.6. X is not selectively sequentially separable. Since S is closed discrete in Q, the set Q \ S is dense i ...
... n ∈ ω such that the set T = Sn ∩ [A]seq is infinite. Since T is a sequence converging to ∞ in X, we finally have [[A]seq ]seq = clX (A). So, X is a sequential space of sequential order 2. Fact 2.6. X is not selectively sequentially separable. Since S is closed discrete in Q, the set Q \ S is dense i ...
DIFFERENTIABLE GROUP ACTIONS ON HOMOTOPY SPHERES. II
... on a neighborhood of AfG and on MG itself. On the other hand, if M and AfG are connected and Af is orientable in the usual sense, then a [/-orientation of Af induces a unique orientation of Af in this sense. A similar statement is true for the fixed point set of each subgroup of G. 5. Given a [/-ori ...
... on a neighborhood of AfG and on MG itself. On the other hand, if M and AfG are connected and Af is orientable in the usual sense, then a [/-orientation of Af induces a unique orientation of Af in this sense. A similar statement is true for the fixed point set of each subgroup of G. 5. Given a [/-ori ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.