Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it $$ d(u,v) := \frac{1}{2^r} \mbox{ with } r := \min\{ n : u_n \ne v_n \}. $$ Here $d(u,v) < \frac{1}{2^n}$ iff $u,v$ coincide in their first $n$ symbols. Regarding to this metric we have:
- $U$ is open iff $U = W\cdot X^{\mathbb N_0}$ where $W$ is any collection of finite sequences (i.e. $W \subseteq X^* := X^0 \cup X^1 \cup X^2 \cup \ldots$) and $w \cdot x$ denotes the concatenation of a finite sequence and an infinite sequence $x$, yielding an infinite sequence
- $U$ is closed iff $U$ has the property: if every prefix of a word $u$ is prefix of some word from $U$ then $u \in U$ (because $u \in \overline U$ iff every prefix of $u$ is a prefix of some word of $U$)
- $U$ is clopen iff $U = W \cdot X^{\mathbb N_0}$ where $W$ is any finite collection of finite sequences
And the open balls are the sets of the form $v \cdot X^{\mathbb N_0}$ for some finite sequence $v$. Now this metric could be extended, for this define $I_n(u,v)$ to be true iff every infix up to a length $n$ of $u$ is contained as infix in $v$ and vice verse, namely $$ \begin{array}{lcl} I_n(u,v) & \Leftrightarrow & \forall i \exists j : u_{i+k} = v_{j+k}, k = 0,\ldots,n-1\\ & & \forall i \exists j : v_{i+k} = u_{j+k}, k = 0,\ldots,n-1 . \end{array} $$ and define $$ d'(u,v) := \max\left\{ d(u,v), \frac{1}{2^r} \right\} \mbox{ with } r := \max\{ n : I_n(u,v) \}. $$ Here $d'(u,v) < \frac{1}{2^n}$ iff $u,v$ coincide in all their infixes (including the first $n$ symbols) up to a length of $n$. Now $d(u,v) \le d'(u,v)$, so that convergence in $d'$ implies convergence in $d$, and every set that is open with regard to $d'$ is open with regard to $d$ (and also for closed sets), so that the topology induced by $d'$ is a refinement of the topology induced by $d$. Now I want to get results about the open sets and the topology induced by $d'$, and their relations. I already know that $d'$ is not complete.
In particular what closed sets with regard to $d$ are open with regard to $d'$ (I already know that not all $d$-closed sets are $d'$-open)? Any suggestions or hints how to attack this problem?
