Is there a distance on $\mathbb{R}$ so that a non-empty subset is open iff its complement is finite?
39010
Metrization of the cofinite topology
101669
Cardinality of the Set of all finite subset of $\mathbb{R}$
111561
Distance of a point to a subset.
158018
Everywhere Super Dense Subset of $\mathbb{R}$
63538
Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
20706
Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)
5251
If $F$ is closed, does there exist a finite measure on $[0,1]$ with support $F$?
127972
Can distance between two closed sets be zero?
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
148992
A finite set is closed
136106
Is every open subset of $ \mathbb{R} $ uncountable?
36631
Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
130297
$A$ be a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$?
44430
Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
94825
Open set as a finite union of closed sets
58007
Proof: Show there is set of $n+1$ points in $\mathbb{R}^n$ such that distance between any two distinct points is $1$?
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$