Could you explain to me how to show 4.1 in detail? Definition. Let (X,T) 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 E X and closed set A CX 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 BC V. A T4-space is any space thatis both T and normal.The most important property of a T1-space is that points are closed.Theorem 4.1. A space (X,T) is T, if and only if every point in X is a closed set.For the topological spaces that you know, it is fun to determine which separationaxioms they satisfy. We will soon ask you to construct a chart listing examples alongthe top and separation properties down the side and in each box answer the questionof whether the example of the column has the property of the row. Here are a few ofthose exercises to warm up with.Exercise 4.2. Let X be a space with the finite complement topology. Show that X is T1.Exercise 4.3. Show that Rstd is Hausdorff.

Name: Could you explain to me how to show 4.1 in detail? Definition. Let (X,
Uploaded: 2024-03-20T06:46:03.000Z
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Description: Could you explain to me how to show 4.1 in detail? Definition. Let (X,T) 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 n