To show that $\mathbb R$ is the only non-empty clopen set in $\mathbb R$
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
141114
Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$
63538
Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
2198
Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
144731
Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
92571
Sets with no asymptotical density over $\mathbb N$
105445
Are the rationals a closed or open set in $\mathbb{R}$?
113460
The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself.
115272
Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
103314
Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty?
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
73130
A simple question about open set
71653
Why is the Cartesian product of a set $A$ and empty set an empty set?
70264
Prove that the inverse image of an open set is open
132382
Showing a set is nowhere dense in $C([0,1])$
10477
Finding the closure of A=$\mathbb Q \cap [0,1]$.