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

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Description: 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

given p ∈ X, and a neighbourhood U of p, there is an open set V ⊂ X suchthat p ∈ V ⊂ U. Consider…

Answered: Theorem 4.8. A topological space X is…

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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.

Advanced Engineering Mathematics

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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

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