Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true? (A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval. (B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded. (C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$. (D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.
Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true?\\
(A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval.\\
(B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded.\\
(C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$.\\
(D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.