Let $(X, \tau)$ be a topological space and $d : X \times X \longrightarrow \mathbb{R}_{\geq 0}$ a continuous function where $X \times X$ has the product topology and $\mathbb{R}_{\geq 0}$ is the set of non-negative real numbers, with the subspace topology of the usual topology of $\mathbb{R}$. Assume that $d^{-1}(0) = \{(x,x) \mid x \in X\}$, and that $d(x,y) \leq d(x,z) + d(y,z)$ for all $x,y,z \in X$. Show the following: (A) $(X, \tau)$ is Hausdorff. (B) The sets $B_{x,\epsilon} := \{y \in X \mid d(x,y) < \epsilon\}$, $0 < \epsilon \in \mathbb{R}$ is the basis for a topology $\tau'$ on $X$. (C) $\tau'$ is coarser than $\tau$ (i.e., every set open in $\tau'$ is open in $\tau$).
Let $(X, \tau)$ be a topological space and $d : X \times X \longrightarrow \mathbb{R}_{\geq 0}$ a continuous function where $X \times X$ has the product topology and $\mathbb{R}_{\geq 0}$ is the set of non-negative real numbers, with the subspace topology of the usual topology of $\mathbb{R}$. Assume that $d^{-1}(0) = \{(x,x) \mid x \in X\}$, and that $d(x,y) \leq d(x,z) + d(y,z)$ for all $x,y,z \in X$. Show the following:\\
(A) $(X, \tau)$ is Hausdorff.\\
(B) The sets $B_{x,\epsilon} := \{y \in X \mid d(x,y) < \epsilon\}$, $0 < \epsilon \in \mathbb{R}$ is the basis for a topology $\tau'$ on $X$.\\
(C) $\tau'$ is coarser than $\tau$ (i.e., every set open in $\tau'$ is open in $\tau$).