Could you explain to me how to show 4.7 in detail? **Theorem 4.7.** 1. A $ T_2 $-space (Hausdorff) is a $ T_1 $-space.2. A $ T_3 $-space (regular and $ T_1 $) is a Hausdorff space, that is, a $ T_2 $-space.3. A $ T_4 $-space (normal and $ T_1 $) is regular and $ T_1 $, that is, a $ T_3 $-space.**Definition.** Let $ (X, \mathcal{T}) $ 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.

Name: Could you explain to me how to show 4.7 in detail? **Theorem 4.7.** 1.
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Description: Could you explain to me how to show 4.7 in detail? **Theorem 4.7.** 1. A $ T_2 $-space (Hausdorff) is a $ T_1 $-space.2. A $ T_3 $-space (regular and $ T_1 $) is a Hausdorff space, that is, a $ T_2 $-space.3. A $ T_4 $-space (normal and \

1) Consider X is a T2 - space. To prove that, X is a T1- space. By the definition of T2- space,…

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Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a Hausdorff space, that is, a T2-space. (3) A T4-space (normal and T¡) is regular and T,, that is, a T3-space.

Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a Hausdorff space, that is, a T2-space. (3) A T4-space (normal and T¡) is regular and T,, that is, a T3-space.

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Could you explain to me how to show 4.7 in detail?

**Theorem 4.7.**

1. A $ T_2 $-space (Hausdorff) is a $ T_1 $-space.

2. A $ T_3 $-space (regular and $ T_1 $) is a Hausdorff space, that is, a $ T_2 $-space.

3. A $ T_4 $-space (normal and $ T_1 $) is regular and $ T_1 $, that is, a $ T_3 $-space.

**Definition.** Let $ (X, \mathcal{T}) $ 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.

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