Question

If \(A\) is a directed set, a subset \(B\) of \(A\) is called cofinal in \(A\) if for each \(\alpha \in A\) there exists \(\beta \in B\) such that \(\beta \gtrsim \alpha\). a. If \(B\) is cofinal in \(A\) and \(\left\langle x_{\alpha}\right\rangle_{\alpha \in A}\) is a net, the inclusion map \(B \rightarrow A\) makes \(\left\langle x_{\beta}\right\rangle_{\beta \in B}\) a subnet of \(\left\langle x_{\alpha}\right\rangle_{\alpha \in A}\). b. If \(\left\langle x_{\alpha}\right\rangle_{\alpha} \in A\) is a net in a topological space, then \(\left\langle x_{\alpha}\right\rangle\) converges to \(x\) iff for every cofinal \(B \subset A\) there is a cofinal \(C \subset B\) such that \(\left\langle x_{\gamma}\right\rangle_{\gamma \in C}\) converges to \(x\).

Short answer

(a) A cofinal subset forms a subnet. (b) Convergence for all cofinal subsets implies net convergence.

Step-by-step solution

Understand the Definition of Cofinality

A subset \( B \) of a directed set \( A \) is cofinal in \( A \) if for each element \( \alpha \) in \( A \), there exists an element \( \beta \) in \( B \) such that \( \beta \gtrsim \alpha \). This means every element of \( A \) is less than or equal to some element of \( B \).

Analyze Part (a) Statement

For part (a), if \( B \) is cofinal in \( A \) and \( \langle x_{\alpha} \rangle_{\alpha \in A} \) is a net, then we consider the inclusion map from \( B \) to \( A \). This inclusion map ensures that there exists a net \( \langle x_{\beta} \rangle_{\beta \in B} \) which is a subnet of the original net \( \langle x_{\alpha} \rangle_{\alpha \in A} \) because \( B \) captures all potential limits of the net through cofinality.

Prove Subnet Existence for Part (a)

Given the inclusion map \( B \to A \) due to \( B \)'s cofinality, every element of \( A \) corresponds to some element in \( B \), ensuring \( \langle x_{\beta} \rangle_{\beta \in B} \) refines \( \langle x_{\alpha} \rangle_{\alpha \in A} \). This refinement matches the definition of a subnet since \( B \) is cofinal.

Analyze Part (b) Statement

Part (b) states that a net \( \langle x_{\alpha} \rangle_{\alpha \in A} \) converges to a point \( x \) in a topological space if, for every cofinal subset \( B \) of \( A \), there exists a cofinal \( C \subset B \) such that the subnet \( \langle x_{\gamma} \rangle_{\gamma \in C} \) converges to \( x \).

Establish Convergence with Cofinal Subsets for Part (b)

To establish the convergence as given in part (b), consider any cofinal subset \( B \) of \( A \). Since \( B \) is cofinal, there exists a cofinal subset \( C \subset B \). If \( \langle x_{\gamma} \rangle_{\gamma \in C} \) converges to \( x \), then the original net \( \langle x_{\alpha} \rangle_{\alpha \in A} \) also converges to \( x \) because nets are characterized by their behavior on cofinal segments.

Key concepts

Directed Set

In the world of mathematics, a directed set is a very useful structure. It consists of a set accompanied by a direction, allowing each pair of elements to have a particular order. More formally, a set \( A \) is directed if it is non-empty and there exists a relation \( \geq \) that satisfies:

• Reflexivity: For every \( \alpha \in A \), \( \alpha \geq \alpha \).
• Transitivity: If \( \alpha \geq \beta \) and \( \beta \geq \gamma \), then \( \alpha \geq \gamma \).
• For any two elements \( \alpha, \beta \in A \), there exists \( \gamma \in A \) such that \( \gamma \geq \alpha \) and \( \gamma \geq \beta \).

These properties make it easier to talk about convergence of sequences and nets, especially in topology. It offers a way to handle direction in non-linear structures.
Directed sets provide a flexible framework, as they do not require total ordering like the standard number line.

Cofinality

Cofinality is a concept closely tied to directed sets. It relates to understanding how a subset can capture the essence of the broader set. Simply put, a subset \( B \) is cofinal in a directed set \( A \) if:

• For every element \( \alpha \in A \), there exists an element \( \beta \in B \) such that \( \beta \geq \alpha \).

In essence, this means that \( B \) is "large" enough to interact with the entirety of \( A \) by having elements that can "match or exceed" all elements of \( A \).
Cofinal subsets are vital in topology, especially when analyzing convergence. They act as key players in the understanding of nets and subnets, ensuring that certain limits are respected in topological spaces.

Topology

Topology is a foundational area of mathematics focused on the properties of space that remain unchanged under continuous transformations. In topology, concepts such as continuity, convergence, and compactness are central.
Nets, which are generalizations of sequences, provide an essential tool for discussing convergence in topological spaces, particularly when the space is non-countable. A net is composed of a directed set which indexes elements that can "move around" in the space according to the topology. In simpler terms, it helps to understand how points "approach" a specific point without the necessity of countability.
Understanding topology involves exploring how these nets navigate space, and concepts like cofinality help determine when a particular navigation results in convergence.

Subnet

The concept of a subnet is a natural extension of the idea of a net. Think of a subnet as a "refinement" of a net.

• Start with a net \( \langle x_{\alpha} \rangle_{\alpha \in A} \) created from a directed set \( A \).
• If you have a cofinal subset \( B \) of \( A \), you can construct a subnet \( \langle x_{\beta} \rangle_{\beta \in B} \).

The inclusion map \( B \rightarrow A \) ensures that every index from \( A \) relates closely to an index in the cofinal subset \( B \). This leads to the subnet refining the net's behavior and capturing its convergence properties.
Subnets are crucial when dealing with complex topological spaces. They help track changes and establish convergence more effectively, acting as a "zoomed-in" lens on specific parts of a net.