Are there any other sets that are both open and closed other than $\emptyset$ and $\Bbb R$?
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
158155
Is set of isolated points of S closed?
44623
Closed set as a countable intersection of open sets
78399
In $T_1$ space, all singleton sets are closed?
94825
Open set as a finite union of closed sets
31212
The existence of the empty set is an axiom of ZFC or not?
138202
Every open set in $\mathbb{R}$ is a countable union of closed sets
44430
Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
72468
Open and closed balls in $C[a,b]$
144731
Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
113460
The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself.
136106
Is every open subset of $ \mathbb{R} $ uncountable?
148992
A finite set is closed
63538
Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
106944
Is the set defined by y=sin(1/x) open or closed?
111538
If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?