1. **Stating the problem:** We need to understand what a $T_1$ space is, give two examples, and then show that a discrete topological space is a $T_3$ space. 2. **Definition of a $T_1$ space:** A topological space is called a $T_1$ space if for every pair of distinct points $x$ and $y$, there exist open sets $U$ and $V$ such that $x \in U$, $y \notin U$, and $y \in V$, $x \notin V$. In other words, singletons are closed sets. 3. **Examples of $T_1$ spaces:** - Example 1: The discrete topology on any set $X$ is $T_1$ because every singleton set $\{x\}$ is open (and hence closed). - Example 2: The standard topology on the real numbers $\mathbb{R}$ is $T_1$ because for any two distinct points, we can find open intervals separating them. 4. **Showing discrete topological space is $T_3$:** - A $T_3$ space (regular Hausdorff) is a $T_1$ space where for any closed set $F$ and point $x \notin F$, there exist disjoint open sets separating $x$ and $F$. - In a discrete space, every set is open and closed. - Given $x \notin F$, since $F$ is closed, $F$ is open as well. - We can take $U = \{x\}$ and $V = F$ which are disjoint open sets separating $x$ and $F$. - Hence, discrete spaces are $T_3$. **Final answer:** A $T_1$ space is one where singletons are closed. Examples include discrete spaces and the real numbers with standard topology. Discrete spaces are $T_3$ because every set is open and closed, allowing separation of points and closed sets by disjoint open neighborhoods.