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Theorem 4.8. A topological space X is regular if and only if for each point p in X and ppen set U containing p there exists an open set V such that p E V and V U.

Theorem 4.8. A topological space X is regular if and only if for each point p in X and ppen set U containing p there exists an open set V such that p E V and V U.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Could you explain how to show 4.8 in detail?

**Theorem 4.8.** A topological space \( X \) is regular if and only if for each point \( p \) in \( X \) and open set \( U \) containing \( p \) there exists an open set \( V \) such that \( p \in V \) and \( \overline{V} \subset U \). **Theorem 4.9.** A topological space \( X \) is normal if and only if for each closed set \( A \) in \( X \) and open set \( U \) containing \( A \) there exists an open set \( V \) such that \( A \subset V \) and \( \overline{V} \subset U \). **Definition.** Let \( (X, \mathcal{J}) \) be a topological space. 1. \( X \) is a \( T_1 \)-space if and only if for every pair \( x, y \) of distinct points there are open sets \( U, V \) such that \( U \) contains \( x \) but not \( y \), and \( V \) contains \( y \) but not \( x \). 2. \( X \) is Hausdorff, or a \( T_2 \)-space, if and only if for every pair \( x, y \) of distinct points there are disjoint open sets \( U, V \) such that \( x \in U \) and \( y \in V \). 3. \( X \) is regular if and only if for every point \( x \in X \) and closed set \( A \subset X \) not containing \( x \), there are disjoint open sets \( U, V \) such that \( x \in U \) and \( A \subset V \). A \( T_3 \)-space is any space that is both \( T_1 \) and regular. 4. \( X \) is normal if and only if for every pair of disjoint closed sets \( A, B \) in \( X \), there are disjoint open sets \( U, V \) such that \( A \subset U \) and \( B \subset V \). A \( T_4 \)-space is any space that is both \( T_1 \) and normal. **Theorem 4.7
Transcribed Image Text:**Theorem 4.8.** A topological space \( X \) is regular if and only if for each point \( p \) in \( X \) and open set \( U \) containing \( p \) there exists an open set \( V \) such that \( p \in V \) and \( \overline{V} \subset U \). **Theorem 4.9.** A topological space \( X \) is normal if and only if for each closed set \( A \) in \( X \) and open set \( U \) containing \( A \) there exists an open set \( V \) such that \( A \subset V \) and \( \overline{V} \subset U \). **Definition.** Let \( (X, \mathcal{J}) \) be a topological space. 1. \( X \) is a \( T_1 \)-space if and only if for every pair \( x, y \) of distinct points there are open sets \( U, V \) such that \( U \) contains \( x \) but not \( y \), and \( V \) contains \( y \) but not \( x \). 2. \( X \) is Hausdorff, or a \( T_2 \)-space, if and only if for every pair \( x, y \) of distinct points there are disjoint open sets \( U, V \) such that \( x \in U \) and \( y \in V \). 3. \( X \) is regular if and only if for every point \( x \in X \) and closed set \( A \subset X \) not containing \( x \), there are disjoint open sets \( U, V \) such that \( x \in U \) and \( A \subset V \). A \( T_3 \)-space is any space that is both \( T_1 \) and regular. 4. \( X \) is normal if and only if for every pair of disjoint closed sets \( A, B \) in \( X \), there are disjoint open sets \( U, V \) such that \( A \subset U \) and \( B \subset V \). A \( T_4 \)-space is any space that is both \( T_1 \) and normal. **Theorem 4.7
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