Proof that $\left \{ f\in \mathbb{N}^\mathbb{N}\mid f\text{ is injective} \right \}$ is not countable by diagonalization.
124847
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
30049
If $T^m$ is diagonalizable for a $m\in\mathbb N$, then $T$ is diagonalizable.
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
6023
Surjectivity of Composition of Surjective Functions
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
119549
Prove the even integers are countable (by explicitly giving a bijection $\mathbb{N} \to E$)
79056
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
70576
Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
75970
$\mathbb Z[1/2]$ is not finitely generated?
128437
How to prove that a finite set is countable
53168
Surjective, Injective, Bijective Functions from $\mathbb Z$ to itself
28692
Find if set $I$ of all injective functions $\mathbb{N} \rightarrow \mathbb{N}$ is equinumerous to $\mathbb{R}$
93923
No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
136538
How can I show that a bijection between $[0,1)$ and $(0,1)$ cannot be continuous?
53399
Proof of the uncountability of reals using the diagonal argument—problem?
86926
Why isn't this a well ordering of $\{A\subseteq\mathbb N\mid A\text{ is infinite}\}$?
63146
Proof of properties of injective and surjective functions.