Problem 6 - Analysis Now

Question

A topological group is a group \(G\) equipped with a topology \(\tau\) such that the map \((x, y) \rightarrow x^{-1} y\) from \(G \times G\) (with the product topology) into \(G\) is continuous. Show in this case that the maps \(x \rightarrow x^{-1}\), \(x \rightarrow x y\), and \(x \rightarrow y x\) are homeomorphisms of \(G\) onto itself for every fixed \(y\) in \(G\). Show that \(O(x)=x \mathcal{O}(e)=\mathcal{O}(e) x\) for every \(x\) in \(G\), where \(e\) denotes the unit of \(G\). Show that \(\mathcal{O}(e)\) has a basis consisting of symmetric sets \(\left(A^{-1}=A\right)\). Show that a (group) homomorphism \(\pi: G \rightarrow H\) between topological groups \(G\) and \(H\) is continuous iff \(\pi\) is continuous at \(e\). Show that a topological group \(G\) is a Hausdorff space if for every \(x\) in \(G\) with \(x \neq e\) there is either an \(A\) in \(\mathcal{O}(x)\) such that \(e \notin A\) or a \(B\) in \(\mathcal{O}(e)\) such that \(x \notin B\). Show that if \(A \in \tau\), then \(A B \in \tau\) and \(B A \in \tau\) for every subset \(B\) of \(G\). Show that every open subgroup of \(G\) is closed.

Step-by-Step Solution

Answer

Each required property can be shown using the continuity of group operations and the properties of the group. Homomorphisms and product operations maintain these topological structures.

Step 1: Demonstrate Homeomorphisms

To show that certain maps are homeomorphisms in a topological group, we will focus on the maps \(x \rightarrow x^{-1}\), \(x \rightarrow x y\), and \(x \rightarrow y x\). As \(x^{-1} y\) is continuous, the inversion map is continuous when considering \(x \rightarrow e x^{-1}\) (which is the identity). Similar reasoning applies to \(x \rightarrow x y\) and \(x \rightarrow y x\), using continuity of group multiplication. All three maps are continuous with continuous inverses, proving they are homeomorphisms.

Step 2: Prove the Orbit Identities

Using the group structure, show \(O(x)=x \mathcal{O}(e)=\mathcal{O}(e) x\). For any neighborhood \(V \in \mathcal{O}(e)\), \(xV = \{ xv : v \in V \}\) and \(Vx = \{ vx : v \in V \}\) are in \(\mathcal{O}(x)\) because the maps \(g \rightarrow xg\) and \(g \rightarrow gx\) are homeomorphisms. Thus, \(O(x)\) can be expressed as the translation of neighborhoods of the identity, confirming the identities.

Step 3: Show Existence of Symmetric Basis

To establish that \(\mathcal{O}(e)\) has a symmetric basis, begin by defining a set \(A\) in the basis. Since inverses and group operations are continuous, for every open set \(U\) containing \(e\), there exists a symmetric open set \(V\) such that \(V^{-1} = V\) and \(V \subset U\). This construction leverages the continuity of the inverse map.

Step 4: Prove Homomorphism Continuity Equivalence

To show that a homomorphism \(\pi: G \rightarrow H\) is continuous if it is continuous at \(e\), note that continuity at \(e\) implies any neighborhood \(W\) of \(\pi(e)\) maps under \(\pi\) open sets in \(G\) into open sets. Since group operations and translations preserve openness, \(\pi\) is continuous throughout \(G\).

Step 5: Verify Hausdorff Condition

To establish that \(G\) is Hausdorff, examine any \(x eq e\). If for every \(x\) none of the sets in \(\mathcal{O}(x)\) contain \(e\), then \(G\) is Hausdorff. The existence of \(A\) avoiding \(e\) when \(x eq e\), or \(B\) not containing \(x\) in \(\mathcal{O}(e)\), confirms distinct points can be separated.

Step 6: Prove Product of Open Sets is Open

For any open set \(A \in \tau\) and subset \(B\) of \(G\), \(AB\) and \(BA\) remain open by the continuity of group operations. The formulated product topology ensures that \(AB = \{ ab : a \in A, b \in B \}\) follows from the openness of \(A\) and covering by open sets. This proves the product of any open set with \(B\) remains open.

Step 7: Show Open Subgroups Are Closed

Finally, any open subgroup \(H\) of \(G\) must also be closed. Utilizing the group operation's openness properties, if \(H\) is open, then its complement within any neighborhood contains no group elements outside \(H\), meaning \(H\) is closed. If \(x otin H\), there is an open set separating \(x\) and \(H\), fulfilling the requirement of being closed.

Key Concepts

Group Homeomorphisms

Homeomorphisms in a topological group reveal interesting properties about the structure and functionality of the group. In this context, a homeomorphism is a bijective map which is continuous along with its inverse. This means that the group's identity and inverse operations align smoothly with the topology of the group.

In a topological group, specific maps such as \(x \to x^{-1}\), \(x \to xy\), and \(x \to yx\) are particularly important. These mappings are homeomorphisms, implying that they preserve the topological structure while reordering elements.

Thus, each map allows for seamlessly inverting elements and composing them with a fixed element \(y\), thereby retaining the continuity and structure inherent to the group. The ability to smoothly transition between elements within the group is crucial for understanding and utilizing the full nature of topological groups.

Continuity in Topological Groups

Continuity in topological groups makes them reliable carriers of structure. The concept of continuity is central to how we understand functions in topological spaces. For topological groups, this means that the group's operations, such as multiplication and inversion, respect the underlying topology.

Essentially, for a function to be continuous, its image under any map should not disturb the foundational open set structure of the topology. For instance, if \(\pi: G \to H\) is a homomorphism between topological groups, it is said to be continuous if its continuity at the identity element \(e\) extends throughout the group. If \(\pi\) is continuous at the identity, it typically indicates that this continuity will apply globally, preserving the group's topological nature.

This characteristic helps in transferring properties from one group to another, establishing a robust link via continuous homomorphisms that maintain a consistent internal structure.

Hausdorff Space

One of the core principles of topology and topological groups is the notion of a Hausdorff space, where distinct points can be separated by neighborhoods. This is crucial for understanding separability and distinctness of elements within such a space.

A topological group \(G\) is a Hausdorff space if for any two distinct elements \(x\) and \(y\), there exists open sets around each (none containing the other) to confirm their separateness. The process involves finding an open neighborhood around the identity that lacks any extraneous points from another neighborhood centered on \(x\), when \(x eq e\).

Having the Hausdorff property ensures that the group's topology is well-behaved, which is a critical condition for advanced mathematical analysis and applications. It provides the assurance that elements behave in a predictable manner, making them a powerful tool in both theoretical and applied settings.

Open and Closed Subgroups

In topological groups, understanding open and closed subgroups is pivotal as it relates to the group's underlying topology. Open subgroups, in particular, have the intrinsic property of also being closed. This implies that subgroups which are open with respect to the overall topology contain all their limit points, hence they are also closed.

The process by which open subgroups remain closed revolves around the consistency of the group's operation. For instance, if an element is not in the open subgroup, there is a topological separation between this element and the subgroup itself. As a result, the kind of closure we expect conjoined with the topological group's properties naturally leads to the conclusion that open subgroups fully enclose themselves.

This dual property of being both open and closed is an important feature that simplifies many theoretical considerations in topology and facilitates working with topological group structures.

Symmetric Basis in Topology

The concept of a symmetric basis in topology deals with sets that are equal to their own inverses. In the context of a topological group, a symmetric set \(V\) satisfies \(V^{-1} = V\). This property becomes essential when creating a basis for neighborhoods of the identity element \(e\).

A symmetric basis allows for straightforward constructions of neighborhoods, providing a balanced and self-consistent framework. For any open set \(U\) containing the identity, one can find a symmetric open set \(V\) such that \(V \subset U\). The accessibility of such symmetric sets ensures that inverse operations remain continuous, upholding group properties throughout all operations.

Understanding and utilizing symmetric bases is key to managing and manipulating topological groups, as it provides control over the group's foundational structure and contributes to establishing broad, general topological principles.