Let $(X, d)$ be a metric space. Choose the correct statement(s) from below: (A) There exists a metric $\tilde{d}$ on $X$ such that $d$ and $\tilde{d}$ define the same topology and such that $\tilde{d}$ is bounded (i.e., there exists a real number $M$ such that $\tilde{d}(x,y) < M$ for all $x,y \in X$). (B) Every closed subset of $X$ that is bounded with respect to $d$ is compact; (C) $X$ is connected; (D) For every $x \in X$, there exists $y \in X$ such that $d(x,y)$ is a non-zero rational number.
Let $(X, d)$ be a metric space. Choose the correct statement(s) from below:\\
(A) There exists a metric $\tilde{d}$ on $X$ such that $d$ and $\tilde{d}$ define the same topology and such that $\tilde{d}$ is bounded (i.e., there exists a real number $M$ such that $\tilde{d}(x,y) < M$ for all $x,y \in X$).\\
(B) Every closed subset of $X$ that is bounded with respect to $d$ is compact;\\
(C) $X$ is connected;\\
(D) For every $x \in X$, there exists $y \in X$ such that $d(x,y)$ is a non-zero rational number.