Looking at the metric space C[-1,1] with the max metric.
Show that only one of the following subgroups is open.
How many of them are closed?
$A = \{f \in C[-1,1] : f(x)<1 \,\,\,\forall x \in [-1,0);\, f(x)<0 \,\,\, \forall x \in [0,1]\}$
$B = \{f \in C[-1,1] : f(x)<1 \,\,\,\forall x \in [-1,0];\, f(x)<0 \,\,\, \forall x \in (0,1]\}$
$C = \{f \in C[-1,1] : f(x) \leq 1 \,\,\,\forall x \in [-1,0);\, f(x) \leq 0 \,\,\, \forall x \in [0,1]\}$
$D = \{f \in C[-1,1] : f(x) \leq 1 \,\,\,\forall x \in [-1,0];\, f(x) \leq 0 \,\,\, \forall x \in (0,1]\}$
I think $A,B$ are not closed because the sequence $f_n(x)=-\frac{1}{n}$ when $n\rightarrow \infty$ will converge to $f(x)=0$, which is not in either of them.
I still have no idea as for $C,D$ being closed or not, and which of the subgroups is open.
Thanks.