What is discrete group?

A discrete group is a topological group with the discrete topology. Often in practice, discrete groups arise as discrete subgroups of continuous Lie groups acting on a geometric space. For example, is a discrete group, realized as a subgroup of the special linear group.

How do you define a group topology?

Such a topology is said to be compatible with the group operations and is called a group topology. The product map is continuous if and only if for any x, y ∈ G and any neighborhood W of xy in G, there exist neighborhoods U of x and V of y in G such that U ⋅ V ⊆ W, where U ⋅ V := {u ⋅ v : u ∈ U, v ∈ V}.

What is discrete topological space?

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set.

Is every topological group a Lie group?

Every locally compact and locally contractible topological group is a Lie group (Hofmann-Neeb arXiv:math/0609684).

Are groups discrete?

every subgroup of a discrete group is discrete. every quotient of a discrete group is discrete. the product of a finite number of discrete groups is discrete. a discrete group is compact if and only if it is finite.

Are all continuous groups Lie groups?

Continuous coordinate transformations in physics often form Lie groups, in particular the set of all continous rotations of a coordinate frame, SO(3). All of this section so far, in fact, leads to this one conclusion.

Is every topological group normal?

Every topological group G is regular, and if G is T0, then G is T3. (or Tychonoff). Theorem A topological group G is metrizable if and only if G is T0 and first countable (i.e. every point has a countable local basis).

What sets are closed in the discrete topology?

In the discrete topology all subsets of S are both open and closed. A set is defined as closed if its complement with respect to S is open; i.e., C is closed if the set of all elements of S that are not in C, (S-C), is open.

Is a discrete set closed?

Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this.

Is a vector space a topological space?

A topological vector space is a vector space (an algebraic structure) which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

Are groups closed under their operation?

Formal Definition of a Group. A group is a set G, combined with an operation *, such that: The operation is associative. The group is closed under the operation.

How is the empty set open in topology?

Since a function that maps the entire space onto a single point is always continuous, the empty set is open. Take an open set which does not contain the single point. Its inverse image is the empty set. Above is a proof for the definition, however, empty set is open by the definition of a topology.

What is open set in terms of topology?

In mathematics, particularly in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

What is continuous map in topology?

In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.

What is a point set topology?

Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or “closeness” on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness.