The first group axiom says: if you combine any two elements from the set, the result is also in the set. The operation never takes you outside.

This sounds obvious, but it rules out a lot of things.

Where closure holds

The integers under addition. Pick any two integers, add them — you get an integer. $3 + 7 = 10$. $(-4) + 9 = 5$. Always an integer. Closure holds.

Permutations under composition. Compose any two permutations of ${1,2,3}$ — you get another permutation. The result still sends each element somewhere, and no two go to the same place. Closure holds.

Invertible matrices under multiplication. Multiply two invertible $n \times n$ matrices — the result is invertible ($\det(AB) = \det(A)\det(B) \neq 0$). Closure holds.

Where closure fails

The natural numbers ${1, 2, 3, \ldots}$ under subtraction. $3 - 5 = -2$, which isn’t a natural number. The operation takes you outside the set. Not closed.

The odd numbers under addition. $3 + 5 = 8$, which is even. Adding two odd numbers always gives an even number. Not closed.

The integers under division. $3 \div 2 = 1.5$, which isn’t an integer. Not closed.

The key insight

Closure is what makes a group self-contained. You can compose operations freely without ever needing to worry about landing somewhere unexpected. Every sequence of operations stays within the same universe.