Let $[0,1]^\mathbb{N}$ be the set of sequences $(x_1,x_2, \cdots)$ where $x_i \in [0,1]$ for all $i \in \mathbb{N}$. We endow $[0,1]^\mathbb{N}$ a metric topology by defining $$ d(x,y) = \sum_i \frac{1}{2^i} |x_i - y_i|. $$ Let $A \subset [0,1]^\mathbb{N}$ be an open topological ball that includes the zero sequence $0 \in A$. Let $B_d(0,\epsilon) \subset A$ be an open ball on $A$ i.e. $$ B_d(0,\epsilon) =\{x \in A \, | \, d(0,x) < \epsilon\} $$ I have two questions:
1) Is the set $B_d(0,\epsilon)$ path-connected for a small $\epsilon>0$ ?
2) If $B_d(0,\epsilon)$ is path-connected for some $\epsilon >0$. Let $u \in (0,1)^k$ be a fixed vector. We define $C_d(u,\epsilon) \subset B_d(0,\epsilon)$ by fixing the $k$-first coordinates of $x \in B_d(0,\epsilon)$ to u, i.e. $$ C_d(u,\epsilon) =\{x \in B_d(0,\epsilon) \, | \, x_1=u_1, x_2=u_2, \cdots x_k=u_k\}. $$ Is $C_d(u,\epsilon)$ path-connected?
Thanks in advance.
The answer to both questions is yes, they are path connected. Both sets are convex so $t \to tx+(1-t)y$ is a path between x and y. Convexity of $B_d (0,\epsilon)$ and $C_d (u,\epsilon)$ can be verified easily from the definition of the metric $d$.