What is topology?
... Euclidean space • Note that in the topological study of knots, the ends are joined, as opposed to the traditional rope with 2 ends. ...
... Euclidean space • Note that in the topological study of knots, the ends are joined, as opposed to the traditional rope with 2 ends. ...
Topology
... Euclidean space • Note that in the topological study of knots, the ends are joined, as opposed to the traditional rope with 2 ends. ...
... Euclidean space • Note that in the topological study of knots, the ends are joined, as opposed to the traditional rope with 2 ends. ...
THE UNIVERSAL MINIMAL SPACE FOR GROUPS OF
... Following Pestov’s work Uspenskij has shown in [Usp00] that the action of a topological group G on its universal minimal system M (G) (with card M (G) ≥ 3) is never 3-transitive so that, e.g., for manifolds X of dimension > 1 as well as for X = Q, the Hilbert cube, and X = K, the Cantor set, M (G) c ...
... Following Pestov’s work Uspenskij has shown in [Usp00] that the action of a topological group G on its universal minimal system M (G) (with card M (G) ≥ 3) is never 3-transitive so that, e.g., for manifolds X of dimension > 1 as well as for X = Q, the Hilbert cube, and X = K, the Cantor set, M (G) c ...
18. Fibre products of schemes The main result of this section is
... the most trivial examples) and it turns out that there is an appropriate condition for schemes (and in fact morphisms of schemes) which is a replacement for the Hausdorff condition for topological spaces. Finally we turn to the problem of glueing morphisms, which is the ...
... the most trivial examples) and it turns out that there is an appropriate condition for schemes (and in fact morphisms of schemes) which is a replacement for the Hausdorff condition for topological spaces. Finally we turn to the problem of glueing morphisms, which is the ...
Chap 0
... K ⇥ y0 where K is a compact subset of X and y0 2 Y then there are open subsets U ⇢ X and V ⇢ Y so that K ⇢ U, y0 2 V and U ⇥ V ✓ W . Proof. For each point x 2 K, W is a nbh of (x, y0 ). So, there exist open nbhs Ux ✓ X of x and Vx ✓ Y of y0 so that Ux ⇥ Vx ✓ W . The open sets Ux form a covering of ...
... K ⇥ y0 where K is a compact subset of X and y0 2 Y then there are open subsets U ⇢ X and V ⇢ Y so that K ⇢ U, y0 2 V and U ⇥ V ✓ W . Proof. For each point x 2 K, W is a nbh of (x, y0 ). So, there exist open nbhs Ux ✓ X of x and Vx ✓ Y of y0 so that Ux ⇥ Vx ✓ W . The open sets Ux form a covering of ...
2. Examples of Groups 2.1. Some infinite abelian groups. It is easy to
... Since rigid motions are special kinds of bijections, for every solid S, the set of all rigid motions of S together with composition (as operation) is a group. In this course we will investigate in depth the groups of rigid motions of the five Platonic solids, which are tetrahedron, cube, octahedron, ...
... Since rigid motions are special kinds of bijections, for every solid S, the set of all rigid motions of S together with composition (as operation) is a group. In this course we will investigate in depth the groups of rigid motions of the five Platonic solids, which are tetrahedron, cube, octahedron, ...
Let T be a locally finite rooted tree and G < Iso(T) be a
... 2. Let T be a locally finite rooted tree and G be a closed subgroup of Iso(T) with a small number of isometry types. Then for every m∈ω and h∈G there is some n∈ω and g∈G \ker πn such that g ker πn ⊆ h ker πn and g ker πn consists of isometries of the same type. If the set of all non-diagonal pair of ...
... 2. Let T be a locally finite rooted tree and G be a closed subgroup of Iso(T) with a small number of isometry types. Then for every m∈ω and h∈G there is some n∈ω and g∈G \ker πn such that g ker πn ⊆ h ker πn and g ker πn consists of isometries of the same type. If the set of all non-diagonal pair of ...
2.4 Points on modular curves parameterize elliptic curves with extra
... 2.5 Genus formulas Let N be a positive integer. The aim of this section is to describe the idea behind computing the genus of X0 (N ), X1 (N ), and X(N ), and to give (without proof) formulas for these genera. The groups Γ0 (1), Γ1 (1), and Γ(1) are all equal to SL2 (Z), so X0 (1) = X1 (1) = X(1) = ...
... 2.5 Genus formulas Let N be a positive integer. The aim of this section is to describe the idea behind computing the genus of X0 (N ), X1 (N ), and X(N ), and to give (without proof) formulas for these genera. The groups Γ0 (1), Γ1 (1), and Γ(1) are all equal to SL2 (Z), so X0 (1) = X1 (1) = X(1) = ...
On a theorem of Jaworowski on locally equivariant contractible spaces
... When the group action is trivial the concept of G-LC agrees with the local contractibility (LC). Next, with respect to simplicity, an example of a G-LC-space is a linear G-space. Since the G-LC property can be easily detected and since it is well-known that G-ANE ⊂ G-LC, there is an interesting ques ...
... When the group action is trivial the concept of G-LC agrees with the local contractibility (LC). Next, with respect to simplicity, an example of a G-LC-space is a linear G-space. Since the G-LC property can be easily detected and since it is well-known that G-ANE ⊂ G-LC, there is an interesting ques ...
Coordinate Geometry
... Summary of Methods for Coordinate Geometry Proofs To prove that a figure is isosceles or equilateral, use: 1. The distance formula to show equal lengths. To prove that a triangle is a right triangle, use: 1. The distance formula to verify the Pythagorean Theorem; or 2. The slope formula to show that ...
... Summary of Methods for Coordinate Geometry Proofs To prove that a figure is isosceles or equilateral, use: 1. The distance formula to show equal lengths. To prove that a triangle is a right triangle, use: 1. The distance formula to verify the Pythagorean Theorem; or 2. The slope formula to show that ...
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.