Prove that if $A$ is both open and closed, $A=\mathbb R$.
116009
Proving that if a set is both open and closed then it is equal to the real numbers
70552
Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$.
16654
Unions and Intersections of Open Sets are Open
47
Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
105445
Are the rationals a closed or open set in $\mathbb{R}$?
50344
What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?
82825
Proving set of interiors is open
33014
Bijection between an open and a closed interval
2198
Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
47680
Prove that the union of the interior of a set and the boundary of the set is the closure of the set
13026
Prove that if AB=I_m, then rk(A)=rk(B).
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
110957
No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
13162
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
1707
How does one prove that addition on $\mathbb{R}$ is associative?
46286
Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.