# General Topology – Open / Closed properties of the Word metric

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?