Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below. (A) $X$ is not locally connected but path connected. (B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ). (C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open. (D) The closed and bounded subsets of $X$ are compact.
Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below.\\
(A) $X$ is not locally connected but path connected.\\
(B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ).\\
(C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open.\\
(D) The closed and bounded subsets of $X$ are compact.