Could you explain to me how to show 4.2 in detail? **Definition.** Let $(X, \mathscr{T})$ 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.The most important property of a $T_1$-space is that points are closed.**Theorem 4.1.** A space $(X, \mathscr{T})$ is $T_1$ 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 separation axioms they satisfy. We will soon ask you to construct a chart listing examples along the top and separation properties down the side and in each box answer the question of whether the example of the column has the property of the row. Here are a few of those exercises to warm up with.**Exercise 4.2.** Let $X$ be a space with

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Description: Could you explain to me how to show 4.2 in detail? **Definition.** Let $(X, \mathscr{T})$ 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 $U$

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Answered: Exercise 4.2. Let X be a space with the…

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Exercise 4.2. Let X be a space with the finite complement topology. Show that X is T1.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Could you explain to me how to show 4.2 in detail?

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

The most important property of a $T_1$-space is that points are closed.

**Theorem 4.1.** A space $(X, \mathscr{T})$ is $T_1$ 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 separation axioms they satisfy. We will soon ask you to construct a chart listing examples along the top and separation properties down the side and in each box answer the question of whether the example of the column has the property of the row. Here are a few of those exercises to warm up with.

**Exercise 4.2.** Let $X$ be a space with

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