prove that if a nonempty subset $S$ of $\mathbb{R}^n$ is both open and closed, then $S=\mathbb{R}^n$
113460
The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself.
99236
Interior of closure of an open set
153934
How to prove that a set R\Z is open
44623
Closed set as a countable intersection of open sets
21114
$S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?
15645
Is convex open set in $\mathbb{R}^n$ is regular?
131983
Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
47975
Closed cone in the euclidean space $\mathbb{R}^n$
129081
Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
126463
Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic?
49220
Open Sets of $\mathbb{R}^1$ and axiom of choice
70264
Prove that the inverse image of an open set is open
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
46917
How to prove that $GL(n,\mathbb R)$ is not connected subset and open subset of$M_n (\mathbb{R})$
86749
Prove that any set of vectors $S \subset \mathbb{R}^n$ containing the zero vector cannot be linearly independent
8015
Compact set plus a closed set is a closed set