Real line with post office metric

A space $X$ with a particular point $p$ satisfying:

• Each point $x\ne p$ is isolated.
• The point $p$ has a countable neighborhood base $\{V_n:n<\omega\}$ such that $V_0=X$, $V_{n+1}\subseteq V_n$ and $|V_n\setminus V_{n+1}|=|\mathbb R|$ for each n, and $\bigcap_{n<\omega}V_n=\{p\}$.

These conditions uniquely determine the space $X$ up to homeomorphism, as shown in Math StackExchange 4833751.

Here are a few examples of spaces satisfying the conditions (see Math StackExchange 4834226).

• The space $X=\mathbb{R}$ with $p=0$. The basic open neighborhoods of $0$ are the Euclidean open intervals around $0$.

• The post office metric space defined as counterexample #139 ("The Post Office Metric") in DOI 10.1007/978-1-4612-6290-9, where $X=\mathbb R^n$ with $n\ge 1$ and $p$ is the origin. The post office metric $d$ on $\mathbb R^n$ is defined by:

• The ordered space $(\mathbb R\times\mathbb Z)\cup\{\infty\}$ with its order topology, where $\mathbb R\times\mathbb Z$ has its lexicographical ordering and $\infty$ is an additional maximum point (and the particular point).