Duncan's space

Let $A$ be the set of all strictly increasing, infinite sequences of positive integers. For any $x\in A$, define $N(n,x)$ as the number of elements of $x$ less than $n$. Let $\delta (x) =\lim_{n\to\infty} \frac{N(n,x)}{n}$. Let $X$ be the subset of $A$ containing precisely the sequences $x$ for which $\delta(x)$ exists. Now for any $x,y\in X$ define $k(x,y)$ to be the smallest index $i$ such that $x_i\neq y_i$. We define a metric $d(x,y)$ on $X$ as

Duncan's Space is the topology on $X$ induced by this metric.

Defined as counterexample #136 ("Duncan's Space") in DOI 10.1007/978-1-4612-6290-9.

Introduced and studied by R. L. Duncan in zbMATH 0085.03002 (https://www.jstor.org/stable/2309919) and zbMATH 0109.03302 (https://www.jstor.org/stable/2309171).