No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
39114
Prove that a set in $\mathbb R^3$ is not an algebraic set
129205
A question about uncountable, dense sets in R
21892
Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
42145
Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
78117
Closed property of nonempty finite set
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
88099
Not $\sigma$-compact set without axiom of choice
149317
An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
21599
$(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
79424
Don't understand proof that interior of a set is open
78300
Showing a set is closed other than giving a counterexample?
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
113915
A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$
15645
Is convex open set in $\mathbb{R}^n$ is regular?