Theorem 4.17. Let X and Y be regular. Then X ×Y is regular. (3) X is regular if and only if for every point x E X and closed set A C X not containing x, there are disjoint open sets U,V such that x E U and A c V. A T3-space is any space that is both T, and regular. Definition. Suppose X and Y are topological spaces. The product topology on the product X x Y is the topology whose basis is all sets of the form U × V, where U is an open set in X and V is an open set in Y.

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Could you explain how to show 4.17 in easiest possible way(in very detail)?

**Theorem 4.17.** Let \( X \) and \( Y \) be regular. Then \( X \times Y \) is regular.

(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.

**Definition.** Suppose \( X \) and \( Y \) are topological spaces. The **product topology** on the product \( X \times Y \) is the topology whose basis is all sets of the form \( U \times V \), where \( U \) is an open set in \( X \) and \( V \) is an open set in \( Y \).
Transcribed Image Text:**Theorem 4.17.** Let \( X \) and \( Y \) be regular. Then \( X \times Y \) is regular. (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. **Definition.** Suppose \( X \) and \( Y \) are topological spaces. The **product topology** on the product \( X \times Y \) is the topology whose basis is all sets of the form \( U \times V \), where \( U \) is an open set in \( X \) and \( V \) is an open set in \( Y \).
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