Metric Spaces: Explore the relationship between $\partial_P S$ and $\partial_{X_{i}} \pi_i(S)$ for $i \in \mathbb N_n.$ Suppose $n \in \mathbb N$ and, for each $i \in  \mathbb N_n$, $(X_i, \tau_i)$ is a non-empty metric space. Suppose $d$ is a conserving metric on $P = \prod_{i=1}^n X_i$. Suppose
$S \subseteq P$. Explore the relationship between $\partial_P S$ and $\partial_{X_i} \pi_i(S)$ for $i ∈ \mathbb  N_n.$
Note that $\partial_P S$ refers to the boundary of $S$ in $P$. $\pi_i(S)$ refers to the $i$th natural projection of $S$ into $X_i$.
$\prod_{i=1}^n X_i$ refers to the product tupple.
The definition of a conserving metric is attached at the bottom
Attempt:


$\partial_P S= \{x \in P \mid dist(x,S)=0=dist(x, S^C) \} \qquad \qquad(1)$.
$\partial_{X_i} \pi_i(S) = \{x_i \in X_i \mid dist(~x_i, ~\pi_i(S)~)=0=dist(~x_i, ~\pi_i^c(S)~)\} \qquad \qquad  (2)$.


Also, since $d$ is conserving : 


$\max \{\tau_i(a_i,b_i) \mid i \in N_n \} \le d(a,b) \le \sum_{i=1}^n \tau_i(a_i,b_i) \qquad \qquad (3)$


By definition: $dist (x,S)= \inf \{d(x,s) \mid s \in S\}$. Therefore, by the definition of infimum, we have, 


$\forall~ r >0, \exists \ s_r \in S \mid d(x,s_r)<dist (x,S)+r \qquad\qquad\qquad(4)$


Now, suppose $x \in \partial_P S$. Then, $x$ satisfies $(1)$.
Then, due to inequality $(3)$ and $(4)$, we have : 
$~\tau_i(x_i, ~\pi_i~(S)~)=0=~\tau_i(x_i, ~\pi_i^c~(S)~) \qquad \forall \ i \in  \mathbb N_n$
This clearly means that whenever $x$ belongs to $\partial_P S,$ then $x_i ~\in~ \partial_{X_i} \pi_i(S) \quad \forall~ i\in N_n$.
Similarly, we can prove the other way around as well.


However, my textbook Metric Spaces by Michael Searcoid presents the solution as below. The solution is a contradiction to our above analysis. Could anyone please help me understand if my analysis is wrong or in an unlikely situation, the solution could be completely wrong? Thank you very much!













_Proving the other way around


Suppose $x_i \in \partial_{X_i} \pi_i(S)~~\forall~~i \in \mathbb N~$ where $~x_i \in X_i$.
$\implies dist(x_i, \pi_i(S))=0=dist(x_i~,~(~ \pi_i(S)~)^c)~\forall~i \in \mathbb N$
$dist(x_i, \pi_i(S))=0~~\forall~~i \in \mathbb N$
$\implies \forall~ r>0, ~ \exists~ s_{ir} \in \pi_i(S)~|~\tau_i(x_i,s_{ir}) < \frac {r}{n}$
By definition of a conserving metric: we have $d(a,b) \le \sum_{i=1}^n \tau_i(a_i,b_i)$
$\implies d(x,s_r)<r~$ where $~s_r \in S$ as each $s_{ir} \in \pi_i(S)$ and $x \in P$ as $x_i \in X_i$.
$\implies dist(x,S)=0 \quad \quad \quad (1)$.
Similarly: $dist(x_i~,~(~ \pi_i(S)~)^c)=0~\forall~i \in \mathbb N$
$\implies \forall~ r>0, ~ \exists~ s'_{icr} \in (~\pi_i(S)~)^c~|~\tau_i(x_i,~s'_{icr}~) < \frac {r}{n}~\forall~ i \in \mathbb N_n~$
where each $~s'_{icr} \in (~ \pi_i(S)~)^c$.
By definition of a conserving metric: we have $d(x,s'_{cr}) \le \sum_{i=1}^n \tau_i(x_i,~s'_{icr}~)$
$\implies d(x,s'_{cr}) \le r~$ where $~s'_{cr} \in~ S^c$ as each $~s'_{icr} \in (~ \pi_i(S)~)^c$.
$\implies dist(x,S^c)=0 \quad \quad \quad (2)$
from $(1),(2)$, we have $ dist(x,S^c)=0= dist(x,S) $
which means $x \in \partial_P S$.
What could be the error which has made in this proof here? Thank you!
 A: 
Explore the relationship between $\partial_P S$ and $\partial_{X_i} \pi_i(S)$ for $i \in \mathbb  N_n.$

As I have understood, you question is only about a particular problem. 
Also,  as I have understood given a point $t\in P$ by $t_i$ you denote $\pi_i(t)$.
The error is here:

$~\tau_i(x_i, ~\pi_i~(S)~)=0=~\tau_i(x_i, ~\pi_i^c~(S)~)$

Whereas the first equality follows from the fact that $\tau_i(x_i, s_{ri})<r$, for the second the situation is different, because if $s_r^c$ be a point in $S^c$ such that $d(x,s^c_r)<r$ then $s_{ri}$ is contained in $\pi_i(S^c)$, but not necessarily in $\pi^c_i(S)$ (which I understood as $\pi_i(S)^c$). The later set may be empty.
As I understood, this possibility is illustrated by the second part of Q 3.11.
The other way around is even worse. Let $n=2$ and $x_i\in\partial_PS$ for each $i$. 
Then for each positive $r$ there exist points $(s_{r1}, s_{r2})$ and $(s’_{r1}, s’_{r2})\in P$ such that $\tau_1(x_1, s_{r1})<r$ and $\tau_2(x_2, s’_{r2})<r$. But this does not imply that there exists a point $(s^*_{r1},s^*_{r2})\in P$ such that $\tau_i(x_i, s^*_{ri})<r$ for each $i$. This possibility is illustrated by the first part of Q 3.11.
