Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties:
$d(x,y)\ge 0$ for all $x$, $y \in X$ and
$d(x,x) = 0$ for all $x$, $y \in X$.
The $d$-ball of radius $r$ centered at $x$ is defined as $B_r(x) = \{y \in X : d(x,y)< r\}$. We'll say the topological space $(X,T)$ is $d$-metrizable if there is a function $d$ such that $U$ is in $T$ if and only if for all $x$ in $U$ there exists a $d$-ball centered at $x$ which is contained in $U$.
I want to show by example that a space can be "$d$-metrizable", meaning it has the properties above, and separable but not second-countable.
Note: I realize that a regular metric space which is separable is second-countable. This one, however, is not and I need an example to show that.
I also need to show that a spaces which are d-metrizable are sequential. It's a long question, I realize. Thank you for your help.
Define
$$d:\Bbb R\times\Bbb R\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases} x-y,&\text{if }x\ge y\\ 1,&\text{if }x<y\;. \end{cases}$$
For $\epsilon\le 1$ we have
$$\{y\in\Bbb R:d(x,y)<\epsilon\}=[x,x+\epsilon)\;,$$
so this generates the topology of the Sorgenfrey line, also known as the lower-limit topology. This space is separable, since the rationals are dense, but it’s not second countable. (By the way, $d$ even satisfies the triangle inequality.)