1. Let's state the problem: We want to explore an example related to the $T_0$ (Kolmogorov) separation axiom using usual topology, interval topology, or cofinite topology. 2. Recall the $T_0$ separation axiom definition: A topological space $(X, \tau)$ is $T_0$ if for every pair of distinct points $x,y \in X$, there exists an open set that contains one of these points but not the other. 3. Example using the usual topology on $\mathbb{R}$: - Consider the real numbers $\mathbb{R}$ with the usual topology (open intervals). - Take two distinct points $x,y \in \mathbb{R}$, say $x=1$ and $y=2$. - The open interval $(1,1.5)$ contains $x=1$ but not $y=2$. - Similarly, $(1.5,2.5)$ contains $y=2$ but not $x=1$. - Hence, the usual topology on $\mathbb{R}$ is $T_0$. 4. Example using the cofinite topology on a set $X$: - The cofinite topology $\tau$ on $X$ is defined by: open sets are either empty or have finite complement. - Take two distinct points $x,y \in X$. - The set $U = X \setminus \{y\}$ is open since its complement $\{y\}$ is finite. - $U$ contains $x$ but not $y$. - Similarly, $X \setminus \{x\}$ contains $y$ but not $x$. - Thus, the cofinite topology is $T_0$. 5. Summary: Both the usual topology on $\mathbb{R}$ and the cofinite topology on any set $X$ satisfy the $T_0$ separation axiom because for any two distinct points, there is an open set containing one but not the other. Final answer: The usual topology on $\mathbb{R}$ and the cofinite topology on any set $X$ are examples of $T_0$ spaces.