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 \(\mathbb{R}\).

Note that \(\mathbb{B}:=\{p\} \cup \{\{p,q\}: q\in X,q\neq p \}\) is a basis. We check that the topology \(\Omega_{B}\) generated by \(\mathbb{B}\) is the VIP topology on \(X\). Let \(U\) be a subset of \(X\) containing \(p\). If \(x \in U\) then choose \(B=\{p\}\)if \(x=p\), and \(B=\{p,x\}\) otherwise. Note further that if \(p\notin U\)then there is no \(B \in \mathbb{B}\) such that \(B \subseteq U\). This shows that \(\Omega_{\mathbb{B}}\) is precisely the VIP topology on \(X\).

Show that the topology generated by the basis \(\mathbb{B}:={X}\cup\{\{q\}:q\in X,q\neq p \}\) is the outcast topology.

Show that the topological space \(\mathbb{N}\) of positive numbers with topology generated by arithmetic progression basis is Hausdorff.

Let \(X:=\mathbb{R}\). Consider the pairs of bases:

(1)\(\mathbb{B_1}:=\{(a,b) \subseteq \mathbb{R} :a,b \in \mathbb{R} \}\) and \(\mathbb{B_2}:=\{(a,b) \subseteq \mathbb{R} :a,b \in \mathbb{Q} \}\).

(2) \(\mathbb{B_1}:=\{[a,b) \subseteq \mathbb{R} :a,b \in \mathbb{R} \}\) and \(\mathbb{B_2}:=\{[a,b) \subseteq \mathbb{R} :a,b \in \mathbb{Q} \}\).

(3) \(\mathbb{B_1}:=\{[a,b) \subseteq \mathbb{R} :a,b \in \mathbb{R} \}\) and \(\mathbb{B_2}:=\{(a,b) \subseteq \mathbb{R} :a,b \in \mathbb{R} \}\).

Do they generate comparable topologies ? If so then do they generate the same topology ?

([1, H. F¨urstenberg]) Consider \(\mathbb{N}\) with the arithmetic progression topology. Verify the following:

(1) For a prime number p, the basis element {np : n ≥ 1} is closed.

(2) There are infinitely prime numbers.

Show that the open unit disc is open in the product topology on \(\mathbb{R^2}\) . Show further that it is not of form \(U \times V\) for any open subsets \(U\) and \(V\) of \(\mathbb{R}\).