Let (X, d) be a metric space and let \(U\) be a subset of X. Show that \(x\in \overline{U}\), iff for every \(x\in \overline{U}\), there exists a convergent sequence \(\{x_n \} \subseteq U\) such that \(Lim_{n \rightarrow\infty}x_n=x\).

 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 \(\mathbb{R}\).