Which spheres are topological groups?

Zaragoza. 62: 75–79, (2007). It is well known since the 1940’s that S0, S1 and S3 are the only spheres admitting a topological group structure. In this short note we provide an easy and direct proof (without using Lie group theory nor dimension theory) of the fact that S2n does not admit such a structure for any n > 0.

Are topological groups Hausdorff?

A topological group G is called a locally compact group if it is a locally compact space and it is Hausdorff.

Are topological groups Abelian?

In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. The theory of topological groups applies also to TAGs, but more can be done with TAGs. Locally compact TAGs, in particular, are used heavily in harmonic analysis.

How do you describe topology?

Topology can be described either physically or logically. Physical topology means the placement of the elements of the network, including the location of the devices or the layout of the cables. Logical topology maps the flow of data, regardless of the physical layout.

What is the study of topology?

Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called “rubber-sheet geometry” because the objects can be stretched and contracted like rubber, but cannot be broken.

What is a topological structure?

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.

What makes a topology?

More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.