Could you explain how to show 4.9 in detail? Theorem 4.8. A topological space X is regular if and only if for each point p in X andopen set U containing p there exists an open set V such that p E V and V C U.Theorem 4.9. A topological space X is normal if and only if for each closed set A in Xand open set U containing A there exists an open set V such that A C V and V c U.Definition. Let (X,J) be a topological space.(1) X is a T1-space if and only if for every pair x, y of distinct points there are open setsU, V such that U contains x but not y, and V contains y but not x.(2) X is Hausdorff, or a T,-space, if and only if for every pair x, y of distinct pointsthere are disjoint open sets U,V such that x E U and y e V.(3) X is regular if and only if for every point x € X and closed set A C X not containingx, there are disjoint open sets U,V such that x E U and AcV. A T3-space is anyspace that is both T and regular.(4) X is normal if and only if for every pair of disjoint closed sets A, B in X, there aredisjoint open sets U,V such that A C U and B C V. A T4-space is any space thatis both T and normal.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 T,-space.(3) A T4-space (normal and T¡) is regular and T1, that is, a T3-space.

Name: Could you explain how to show 4.9 in detail? Theorem 4.8. A topologica
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Description: Could you explain how to show 4.9 in detail? Theorem 4.8. A topological space X is regular if and only if for each point p in X andopen set U containing p there exists an open set V such that p E V and V C U.Theorem 4.9. A topological space X is norm

Consider X,I be a normal space and U⊂X be an open neighbourhood of a closed set A⊂X, then A⊂U To…

Answered: Theorem 4.9. A topological space X is…

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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 c V and V c U.

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Could you explain how to show 4.9 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 E V and V C 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 C V and V c U.
Definition. Let (X,J) be a topological space.
(1) X is a T1-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,-space, if and only if for every pair x, y of distinct points
there are disjoint open sets U,V such that x E U and y e V.
(3) X is regular if and only if for every point x € X and closed set A C X not containing
x, there are disjoint open sets U,V such that x E U and AcV. A T3-space is any
space that is both T 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 C U and B C V. A T4-space is any space that
is both T and normal.
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 T,-space.
(3) A T4-space (normal and T¡) is regular and T1, that is, a T3-space.

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