2.(5 points) Consider a set $H = \{(x, y) \in \mathbb{R}^2 \mid y \ge 0\}$ and family of subset
$\mathcal{B} := \{
B_r(a, b) \subset \mathbb{R}^2 \mid (a, b) \in H, b > r > 0\}
\cup \{\tilde{B}_r(a, 0) \subset \mathbb{R}^2 \mid (a, 0) \in H, r > 0\}
$
where $B_r(a, b)$ is an open ball at $(a, b)$ with radius $r$ for Euclidean metric of $\mathbb{R}^2$ and
$\tilde{B}_r(a, 0) := \{(x, y) \in H \mid (x - a)^2 + y^2 < r^2, y > 0\} \cup \{(a, 0)\}$.
Is $\mathcal{B}$ a basis of $H$? Justify your answer.
