Describe all topologies on a 2-point set. Give five topologies on a 3-point set

Let $(X,\Omega)$ be a topological space and let $U$ be a subset of $X$. Suppose for every $x\in U$ there exists $U_x\in \Omega$ such that $x \in U_x \subseteq \Omega$. Show that $U$ belongs to $\Omega$.

(Co-countable Topology) For a set $X$, define $\Omega$ to be the collection of subsets $U$ of $X$ such that either $U=\emptyset$ or $X \backslash U$ is countable. Show that $\Omega$ is a topology on $X$.

Let $\Omega$ be the collection of subsets $U$ of $X:=\mathbb{R}$ such that either $X \backslash U= \emptyset$ or $X\backslash U$ is infinite. Show that $\Omega$ is not a topology on $X$.

 Show that the usual topology is finer than the co-finite topology on $\mathbb{R}$.

Show that the usual topology and co-countable topology on $\mathbb{R}$ are not comparable.

The collection $\{(a,b) \subseteq \mathbb{R}:a,b \in \mathbb{Q} \}$ is a basis for a topology on $\mathbb{R}$.

Show that collection of balls (with rational radii) in a metric space forms a basis.

(Arithmetic Progression Basis) Let $X$ be the set of positive integers and consider the collection $\mathbb{B}$ of all arithmetic progressions of positive integers. Then $\mathbb{B}$ is a basis. If $m \in X$ then $B:=\{m+(n-1)p\}$ contains m. Next consider two arithmetic progressions $B_1=\{a_1+(n-1)p_1\}$and $B_2=\{a_2+(n-1)p_2\}$containing an integer $m$. Then$B:=\{m+(n-1)(p)\}$ does the job for $p:=lcm\{p_1,p_2\}$.

Show that the topology $\Omega{_B}$ generated by the basis $\mathbb{B}:=\{(a,b) \subseteq \mathbb{R}:a,b \in \mathbb{Q}\}$ is the usual topology on $\mathbb{R}$.

The collection $\{[a,b) \subseteq \mathbb{R}:a,b \in\mathbb{R}\}$is a basis for a topology on $\mathbb{R}$. The topology generated by it is known as lower limit topology on $\mathbb{R}$.