If every proper closed set of a topological space is compact, then the space itself is compact.
The product of two numbers, each of which is the sum of four squares, is itself a sum of four squares.
A ring with all elements idempotent is commutative.
There are infinitely many pairs of primes that differ exactly by `2`.
Every finite division ring is a field.
A finite graph in which every pair of vertices have precisely one common neighbour contains a vertex that is adjacent to all other vertices.
The number of partitions with odd parts is equal to the number of partitions with distinct parts.
A group whose automorphism group is cyclic is Abelian.
A uniformly continuous function of a uniformly continuous function is uniformly continuous.
A uniformly continuous function of a uniformly continuous function is uniformly continuous.
If a function from the unit interval to itself has a point of period three, then it has points of all positive periods.
The sum of the cubes of two positive integers is never equal to the cube of a third integer.
If every element of a group `G` has order `2`, then every pair of elements of `G` commutes.
The product of two consecutive natural numbers is even.
Every index 2 subgroup of a group is normal.
Every free group is torsion free.
A finite torsion-free group is trivial
Every finite division ring is a field.
Every finite topological space is compact.
Every surjective homomorphism from a finitely generated free group to itself is injective.
Every positive even integer can be written as the sum of two primes.
The square root of an irrational number is irrational.
If the square of a number is even, the number itself is even.
In a finite commutative ring, all prime ideals are maximal.
Every non-identity element of a free group is of infinite order.
For any two relatively prime positive integers $a$ and $b$, every sufficiently large natural number $N$ can be written as a linear combination $ax + by$ of $a$ and $b$, where both $x$ and $y$ are natural numbers.
Every field is a ring.
The set of units in a ring forms a group.
If the direct product of two groups is torsion free then each of the groups is torsion free.
