Non-existence of non-trivial clopen set in $\mathbb R$ without using connectedness
110957
No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
122463
How to show the that a set $A$ nowhere dense is equivalent to the complement of $A$ containing a dense open set?
102048
Closure of a compact set in Regular space $ X $
146560
Is there any uncountably infinite set that does not generate the reals?
44430
Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
21892
Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
72213
Can a $N-1$ rectifiable set be partitioned into countably many connected pieces?
42095
Existence in ZF of a set with countable power set
3896
A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
156520
Intervals are connected and the only connected sets in $\mathbb{R}$
74330
$\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
19954
Perfect set without rational numbers
151719
Connected set in $\mathbb{S}^1$
18334
Equivalence between CH and existence of certain sets in $\mathbb{R}^2$
53429
In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?
92526
Collection of clopen sets form a base for a countable completely regular space
78117
Closed property of nonempty finite set