Question

If \(X\) has the weak topology generated by a family \(F\) of functions, then \(\left\langle x_{\alpha}\right\rangle\) converges to \(x \in X\) iff \(\left\langle f\left(x_{\alpha}\right)\right\rangle\) converges to \(f(x)\) for all \(f \in \mathscr{F}\). (In particular, if \(X=\prod_{i \in I} X_{i}\), then \(x_{\alpha} \rightarrow x\) iff \(\pi_{i}\left(x_{\alpha}\right) \rightarrow \pi_{i}(x)\) for all \(\left.i \in I .\right)\)

Short answer

Sequence converges in weak topology iff it converges under each function in the generating family.

Step-by-step solution

Understanding Weak Topology

The weak topology on a set \(X\) is generated by a family \(F\) of functions, meaning that the topology has the smallest number of open sets such that every function \(f : X \to Y\) for \(f \in F\) remains continuous. This ensures that convergence and continuity properties are tied to this specific family of functions.

Convergence Definition in Topological Spaces

In any topological space, a sequence \(\langle x_\alpha \rangle\) converges to a point \(x\) if for every open set \(U\) containing \(x\), there exists an index \(\alpha_0\) such that for all \(\alpha \geq \alpha_0\), \(x_\alpha \in U\). This ensures that the sequence gets arbitrarily close to \(x\) as it progresses.

Applying Weak Topology

For a sequence \(\langle x_\alpha \rangle\) to converge to \(x\) in the weak topology generated by \(F\), it suffices to show that \(\langle f(x_\alpha) \rangle\) converges to \(f(x)\) for each \(f \in F\). This is due to the definition of the weak topology, where the functions in \(F\) dictate convergence.

Special Case of Product Topology

Consider \(X = \prod_{i \in I} X_i\), the product of topological spaces, with the weak topology given by the projections \(\pi_i : X \to X_i\). A sequence \(x_\alpha\) converges to \(x\) if and only if \(\pi_i(x_\alpha) \to \pi_i(x)\) for every index \(i \in I\). This follows from the fact that the projection functions generate the weak topology in the product space.

Conclusion of Convergence

Therefore, given that \(\langle f(x_\alpha) \rangle \to f(x)\) for all \(f \in F\), which includes the projection functions in the product space, we conclude that \(x_\alpha \to x\) in the weak topology. This connects the behavior of the sequence under the functions with convergence in the topology defined by them.

Key concepts

Convergence in Topology

In topology, convergence describes how a sequence of points behaves as it approaches a particular point. Understanding this concept is crucial in studying how different topologies, or ways of organizing points and their relationships, behave. In any given topological space, to say that a sequence \( \langle x_\alpha \rangle \) converges to a point \( x \), means that for every open set \( U \) containing \( x \), there exists a point in the sequence after which all subsequent points fall within \( U \).
This can be thought of as the sequence getting indefinitely close to \( x \) as it progresses. This definition is fundamental and appears in various contexts in topology, such as the weak topology and the product topology.

This concept forms the bedrock of understanding more advanced topics like weak and product topologies, where additional functions or projections play a role in defining convergence.

Product Topology

The product topology is a way to combine several topological spaces into one larger space, where each component of the space retains its own topological properties. If you have a collection of spaces \( \{ X_i \} \) indexed by some set \( I \), their product, denoted by \( \prod_{i \in I} X_i \), forms a new space. In this product space, a sequence \( x_\alpha \) converges to \( x \) if and only if each of its components also converges in their respective spaces.

In practice, we use projection functions \( \pi_i \), which "project out" the \( i\text{th} \) component from the product space. Convergence in product topology is tested component-wise using these projection functions.

• Convergence is checked by the criterion: \( \pi_i(x_\alpha) \to \pi_i(x) \) for each \( i \in I \).
• This relates directly to how well-behaved functions can interact across the different dimensions of these combined spaces.

A deeper understanding of this comes into play when dealing with weak topologies that involve such products.

Projection Functions

Projection functions are central to understanding both weak and product topologies. A projection function, denoted \( \pi_i \), is a way to "pick out" one coordinate or component from a product of spaces. When you apply a projection function to a point in a product space, it yields the component of the point corresponding to the particular space you are interested in.

These functions are immensely useful when assessing convergence in product topologies. The reason is that convergence in the whole product space equates to checking convergence in each individual space, using the corresponding projection function. Simply put:

• If \( X = \prod_{i \in I} X_i \) is your product space, and \( x_\alpha \) is a sequence in \( X \), then \( x_\alpha \to x \) if \( \pi_i(x_\alpha) \to \pi_i(x) \) for every \( i \).
• Projection functions guarantee the weak topology when used as the family \( F \) of functions that define open sets and continuity in this context.

Thus, projection functions highlight how individual components behave, providing clarity and simplicity when dealing with potentially complex spaces.