Question

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

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 \(\pi\).

Key concepts

Quotient Topology

Quotient topology is a fundamental concept in topological spaces, especially in studies involving continuous functions and equivalence relations. When we have a topological space \(X\) and an equivalence relation \(\sim\), we can "glue" points together to form a new space. This new space, denoted \(X/\sim\), is the set of equivalence classes, where each class is a subset of \(X\) consisting of elements that are equivalent under \(\sim\).
The quotient topology on \(X/\sim\) is defined so that a set \(V\) is open in \(X/\sim\) if and only if its preimage under the quotient map is open in \(X\). The quotient map is continuous by default and also surjective.
Given this setup, the quotient topology captures the idea of continuity and openness from the original space to the new space in a way that respects the equivalence relation.

Discrete Topology

A discrete topology on a set \(X\) means that every subset of \(X\) is open. In topological terms, this is the most refined topology possible—it essentially gives \(X\) the power set topology.
One of the key characteristics of a discrete topology is that it makes the space very "simple" to understand, since every element forms an open set by itself. This means there is no notion of limit points or convergence outside of exact membership.
For any group with a discrete topology, every single subgroup or coset is open, resulting in highly predictable behavior. In such cases, every map from a discrete space becomes continuous, as there are no further constraints on openness or continuity.

Locally Connected Spaces

A space is called locally connected if every point has a neighborhood base consisting of open sets that are connected. This means wherever you "zoom in" on a space, you continue to find smaller connected open subsets around each point.
In a locally connected topology, components become much more controllable and manageable. Open components ensure path continuity within each connected part, making it foundational for studying continuity properties.

• Connected components within such a space are both open and closed.
• This property allows us to delve into complex spaces with the understanding that these components can be consistently related back to the fundamental structure of the space.

As seen in the exercise, the component of the identity in such a group space (\(G_0\)) helps maintain connectivity while ensuring that each coset of \(G / G_0\) retains openness.

Natural Projection Map

The notion of a natural projection map is instrumental in creating quotient spaces. In our context, the map \(\pi: G \to G/G_0\) where each point \(g\) maps to its coset \(gG_0\), establishes a bridge between elements of a group and their respective equivalence classes under group operations.
The natural projection map allows one to jump to this quotient space by maintaining continuity—every point \(g\) is freely sent to its corresponding group of equivalence class \(gG_0\).
Since \(\pi\) is foundational in introducing the topology from the domain \(G\) to the co-domain \(G/G_0\), it serves as a primary tool in transferring properties from one context to another. Given that \(G/G_0\) holds the discrete topology, \(\pi\) ensures that this transition maintains topological regularity.