The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself.
42145
Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
34099
Proving that the empty set is unique
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
156101
Is the convex hull of closed set in $\mathbb R^{n}$ is closed?
82900
Show that Minkowski sum of a closed set and a compact set in $\mathbb{R}^n$ is closed.
136755
Direct proof of empty set being subset of every set
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
149317
An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
95244
Prove that a finite union of closed sets is also closed
150825
Is an empty set equal to another empty set?
78399
In $T_1$ space, all singleton sets are closed?
128844
An empty intersection of decreasing sequence of closed sets
46317
What is an Empty set?
105445
Are the rationals a closed or open set in $\mathbb{R}$?
141806
Prove that a nest of sets has an empty intersection
94825
Open set as a finite union of closed sets
99678
Is a sphere a closed set?