(a) Given a topology $\mathcal{T}$ on $X$, a collection $\mathcal{B} \subseteq \mathcal{T}$ is called a basis for $\mathcal{T}$ if every non-empty set in $\mathcal{T}$ is a union of sets in $\mathcal{B}$. Prove that a collection $\mathcal{B}$ is a basis for some topology if it satisfies:

(i) the union of all sets in $\mathcal{B}$ is $X$;

(ii) if $x \in B_{1} \cap B_{2}$ for two sets $B_{1}$ and $B_{2}$ in $\mathcal{B}$, then there is a set $B \in \mathcal{B}$ with $x \in B \subset B_{1} \cap B_{2}$.

(b) On $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ consider the dictionary order given by

$$\left(a_{1}, b_{1}\right)<\left(a_{2}, b_{2}\right)$$

if $a_{1}<a_{2}$ or if $a_{1}=a_{2}$ and $b_{1}<b_{2}$. Given points $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ let

$$\langle\mathbf{x}, \mathbf{y}\rangle=\left\{\mathbf{z} \in \mathbb{R}^{2}: \mathbf{x}<\mathbf{z}<\mathbf{y}\right\}$$

Show that the sets $\langle\mathbf{x}, \mathbf{y}\rangle$ for $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{2}$ form a basis of a topology.

(c) Show that this topology on $\mathbb{R}^{2}$ does not have a countable basis.