**The question:** Consider the metric space $ chi = ${($ x_1, x_2, …, x_N $}}, with the word metric d (x, y) = number of digits that differ between x and y. Are the following sets open, closed or not?

(I) $ S_1 $ = {x$ in chi $|$ x_i = 0, i = 1, …, 5 $}

(Ii) $ S_2 $ = {x$ in chi $|$ x_ {i + 1} neq x_i, i = 1, …, N-1 $}

My attempt:

Take x $ in S_1 $, Then under r <1, $ B_r (x) = $ {X} $ in S_1 $ Thus, $ S_1 $ is open. But for r <1, $ B_r (x) $, $ B_r (x) / ${x} = $ emptyset $, Thus, $ S_1 $ has no cluster points and contains all cluster points (none) after extension. Thus, $ S_1 $ is also closed.

Likewise, I understand that $ S_2 $ is both open and closed. (I also used the argument $ S_2 $ is finally to show that it was closed because it should only have two elements: $ x_1 $ = 101010 … and $ x_2 $ 010101 …)

Is my logic flawed or was it a problem with how the question was formulated (that open and closed is not an option)? Also more general (not part of the problem), but is every subset of this area $ chi $ both open and closed?