LECTURE NOTES IN TOPOLOGICAL GROUPS 1
... (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the group type: (a) (R, τs ) the Sorgenfrey line. (b) X := {x ∈ R2 : ||x|| = 5}. (c) X := {x ∈ R3 : ||x|| < 5}. (d) The integers Z with the cofinit ...
... (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the group type: (a) (R, τs ) the Sorgenfrey line. (b) X := {x ∈ R2 : ||x|| = 5}. (c) X := {x ∈ R3 : ||x|| < 5}. (d) The integers Z with the cofinit ...
Lines on Projective Hypersurfaces
... Σkp is a cone with vertex p whose underlying space is the union of all lines in Pn passing through p and having contact to order k with X at p, and Σdp is exactly the cone of lines on X passing through p. Since Σkp is the intersection of k hypersurfaces in Pn , its expected dimension is n − k. The n ...
... Σkp is a cone with vertex p whose underlying space is the union of all lines in Pn passing through p and having contact to order k with X at p, and Σdp is exactly the cone of lines on X passing through p. Since Σkp is the intersection of k hypersurfaces in Pn , its expected dimension is n − k. The n ...
A refined HVZ-theorem for asymptotically homogeneous
... A BSTRACT. We study algebras associated to N -body type Hamiltonians with interactions that are asymptotically homogeneous at infinity on a finite dimensional, vector real space X. More precisely, let Y ⊂ X be a linear subspace and vY be a continuous function on X/Y that has uniform homogeneous radi ...
... A BSTRACT. We study algebras associated to N -body type Hamiltonians with interactions that are asymptotically homogeneous at infinity on a finite dimensional, vector real space X. More precisely, let Y ⊂ X be a linear subspace and vY be a continuous function on X/Y that has uniform homogeneous radi ...
KMS states on self-similar groupoid actions
... Suppose that E is a directed graph and G is a groupoid with unit space E 0 . An action of G on the path space E ∗ is a (unit-preserving) groupoid homomorphism φ : G → P(E ∗ ). The action is faithful if φ is one-to-one. If the homomorphism is fixed, we usually write g · µ for φg (µ). This applies in ...
... Suppose that E is a directed graph and G is a groupoid with unit space E 0 . An action of G on the path space E ∗ is a (unit-preserving) groupoid homomorphism φ : G → P(E ∗ ). The action is faithful if φ is one-to-one. If the homomorphism is fixed, we usually write g · µ for φg (µ). This applies in ...
MA3056: Metric Spaces and Topology
... Each x ∈ X appears as the first element in exactly one pair from Gf . Conversely, if G ⊂ X × Y is any subset that satisfies the above property then G defines a function g : X → Y by setting, for each x ∈ X, g(x) ∈ Y to be the unique element of Y such that (x, g(x)) ∈ G. This is the traditional set t ...
... Each x ∈ X appears as the first element in exactly one pair from Gf . Conversely, if G ⊂ X × Y is any subset that satisfies the above property then G defines a function g : X → Y by setting, for each x ∈ X, g(x) ∈ Y to be the unique element of Y such that (x, g(x)) ∈ G. This is the traditional set t ...
The Topological Version of Fodor`s Theorem
... is not bounded. It follows from our assumptions that X itself is unbounded. We say that X is good if the intersection of any two closed unbounded (in short: c u b) subsets of X is non-empty. We shall denote by C(X) the family of all c u b sets in X. It is easy to see that an ordinal space α is good ...
... is not bounded. It follows from our assumptions that X itself is unbounded. We say that X is good if the intersection of any two closed unbounded (in short: c u b) subsets of X is non-empty. We shall denote by C(X) the family of all c u b sets in X. It is easy to see that an ordinal space α is good ...
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR
... isomorphism and the point is that this cannot be done uniformly for all spaces. Thus, although for a single space we can express its singular homology with coefficients in the above terms, this cannot be done in a way compatible with respect to morphisms of spaces. At the end of this section we will ...
... isomorphism and the point is that this cannot be done uniformly for all spaces. Thus, although for a single space we can express its singular homology with coefficients in the above terms, this cannot be done in a way compatible with respect to morphisms of spaces. At the end of this section we will ...
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.