Chapter 10: Problem 22

If \(G_{0}\) is the component of the identity of a locally connected topological group \(G\), the factor group \(G / G_{0}\) is called the group of components of \(G .\) Show that the group of components is a discrete topological group with respect to the topology induced by the natural projection map \(\pi: g \mapsto g G_{0}\)

Short Answer

Expert verified

The group of components \(G / G_{0}\) is a discrete topological group because in the topology of \(G / G_{0}\) induced by the projection map \(\pi\), every coset is an open set and therefore any subset of \(G/G_{0}\) is open. Consequently, the topology satisfies the definition of a discrete topology.

Step by step solution

Verify the Topology Induced by the Natural Projection Map

Let's first verify the topology induced by the natural projection map \(\pi: g \mapsto g G_{0}\). This map is continuous. For each \(g \in G\) and \(a \in G_{0}\), \(\pi(g a)=g G_{0}=\pi(g)\). Therefore, \(\pi\) is a quotient map and the topology on \(G / G_{0}\) is the quotient topology induced by \(\pi\).

Prove G0 is Open in G

G is locally connected and \(G_{0}\) is connected, because it is the component of the identity, and in locally connected spaces, connected components are open. Thus, \(G_{0}\) is an open subset of G.

Prove Every Coset in G/G0 is Open

Next, we prove every coset in \(G / G_{0}\) is open. Given an arbitrary coset \(g G_{0}\) in \(G / G_{0}\), its preimage under the map \(\pi\) is \(\pi^{-1}(g G_{0})=g G_{0}\), which is open in G by left multiplication in a topological group. An open set maps to an open set, so by definition \(g G_{0}\) is open in \(G / G_{0}\). Therefore, every subset of \(G / G_{0}\) is open.

Show G / G0 is a Discrete Topological Group

According to the definitions, we can see that in the topology of \(G / G_{0}\), every single coset is an open set. This implies that \(G / G_{0}\) has the discrete topology. Therefore, the group of components \(G / G_{0}\) is a discrete topological group under the quotient topology induced by the projection map \(\pi\).