Algebraic Transformation Groups and Algebraic Varieties
... G if the intersection of Ru (H) with any reductive subgroup of G is trivial, or equivalently, if Ru (H) ∩ M g = 1 for all g ∈ G. If H satisfies Matsushima’s Criterion, then, as we have seen in the proof of the lemma, a maximal reductive subgroup M of G acts freely on G/Ru (H). It is easy to see that ...
... G if the intersection of Ru (H) with any reductive subgroup of G is trivial, or equivalently, if Ru (H) ∩ M g = 1 for all g ∈ G. If H satisfies Matsushima’s Criterion, then, as we have seen in the proof of the lemma, a maximal reductive subgroup M of G acts freely on G/Ru (H). It is easy to see that ...
Summer School Topology Midterm
... space. That means that for any x ∈ X there is compact neighborhood of x, i.e. for any x ∈ X, there is a compact subset K containing an open subset U such that x ∈ U ⊆ K. Let ∞ be an element not in X. On Y = X ∪ {∞} consider the topology generated by the open subsets of X and the complements in Y of ...
... space. That means that for any x ∈ X there is compact neighborhood of x, i.e. for any x ∈ X, there is a compact subset K containing an open subset U such that x ∈ U ⊆ K. Let ∞ be an element not in X. On Y = X ∪ {∞} consider the topology generated by the open subsets of X and the complements in Y of ...
§5 Manifolds as topological spaces
... we have plenty of functions: first of all, the standard coordinate functions xi , then polynomials and various other smooth functions of the variables x1 , . . . , xn . By contrast, in the absence of global coordinates on a manifold M n , how one can find a non-trivial smooth function? If, however, ...
... we have plenty of functions: first of all, the standard coordinate functions xi , then polynomials and various other smooth functions of the variables x1 , . . . , xn . By contrast, in the absence of global coordinates on a manifold M n , how one can find a non-trivial smooth function? If, however, ...
Ordered Topological Structures
... as a full bireflective subcategory but fail to possess some of the good properties that ModTop enjoys as a topological category over Ord (and Set). The setting in which we discuss these categories and their functorial interactions is much more general than these introductory remarks may suggest. We ...
... as a full bireflective subcategory but fail to possess some of the good properties that ModTop enjoys as a topological category over Ord (and Set). The setting in which we discuss these categories and their functorial interactions is much more general than these introductory remarks may suggest. We ...
CONVERGENT SEQUENCES IN TOPOLOGICAL SPACES 1
... Then the sequence xn = (−1) converges to 0. In fact it converges to any point in [−2, 2i. Show that the sequence yn = sin n is also convergent. 3. If X is equipped with the trivial topology T = {∅, X} then any sequence in X is convergent and its limit is any point in X. 4. Let X = R be given the fin ...
... Then the sequence xn = (−1) converges to 0. In fact it converges to any point in [−2, 2i. Show that the sequence yn = sin n is also convergent. 3. If X is equipped with the trivial topology T = {∅, X} then any sequence in X is convergent and its limit is any point in X. 4. Let X = R be given the fin ...
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.