How do I show 6.2? Could you explain it in detail? **Definitions and Theorem on Covers and Compactness****Definition.** Let \( A \) be a subset of \( X \) and let \( \mathcal{C} = \{C_\alpha\}_{\alpha \in \lambda} \) be a collection of subsets of \( X \). Then \( \mathcal{C} \) is a **cover** of \( A \) if and only if \( A \subseteq \bigcup_{\alpha \in \lambda} C_\alpha \). The collection \( \mathcal{C} \) is an **open cover** of \( A \) if and only if \( \mathcal{C} \) is a cover of \( A \) and each \( C_\alpha \) is open. A **subcover** \( \mathcal{C}' \) of a cover \( \mathcal{C} \) of \( A \) is a subcollection of \( \mathcal{C} \) whose elements form a cover of \( A \).For instance, the open sets \(\{(-n,n)\}_{n \in \mathbb{N}}\) form an open cover of \( \mathbb{R} \). A subcover of this cover is \(\{(-n,n)\}_{n \geq 5}\), because these sets still cover all of \( \mathbb{R} \).**Definition.** A space \( X \) is **compact** if and only if every open cover of \( X \) has a finite subcover.**Theorem 6.2.** Let \( C \) be a compact subset of \( \mathbb{R}_{\text{std}} \). Then \( C \) has a maximum point, that is, there is a point \( m \in C \) such that for every \( x \in C \), \( x \leq m \).

Name: How do I show 6.2? Could you explain it in detail? **Definitions and T
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: How do I show 6.2? Could you explain it in detail? **Definitions and Theorem on Covers and Compactness****Definition.** Let \( A \) be a subset of \( X \) and let \( \mathcal{C} = \{C_\alpha\}_{\alpha \in \lambda} \) be a collection of subsets of \(