If $X$ is uncountable, then the set of bounded injective functions: $f: X \rightarrow \mathbb{R}$ has empty interior.
104842
Let $X$ be uncountable. Then the set of injective functions $f\colon X \to \mathbb{R}$ has empty interior in $\mathscr{C}(X;\mathbb{R})$
88201
Suppose $X$ is a finite set and $f : X \to X$ is a function. Then $f$ is injective if and only if $f$ is surjective.
15438
do all uncountable sets have same cardinality as real numbers?
7723
Countable basis of function spaces
63678
Let $A$ and $B$ be countable sets. Is there any function $f$ such that a certain condition holds for an uncountable number of functions $g$?
80827
Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology
130981
Co-countable measure on uncountable set
17525
Why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty?
76809
"Some" and "Any"; countables and uncountable
31212
The existence of the empty set is an axiom of ZFC or not?
72750
Uncountable number of mutually disjoint countable sets in $\mathbb R$
32070
Prove that the Interior of the Boundary is Empty
99599
Uncountable and countable
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
22391
Is the set of all pairs of real numbers uncountable?
82371
$f$ is an entire function with Im $f\geq 0$