Examples of "transfer via bijection"

Question

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then it doesn't matter which set of cardinality $\varkappa$ we choose. But sometimes choosing a set which has some additional structure might make proving/disproving whether the set has this property.

I am posting some examples of this method as answers below - I hope they help to clarify what I have in mind.

Do you know of some other interesting examples?

Answer 1

Suppose that $|A|=\aleph_0$. We are asking whether there exists an almost disjoint system of infinite subsets of the set $A$, which has cardinality $\mathfrak c$. Recall that a system $\mathcal S$ is called almost disjoint if $A\cap B$ is finite for any two sets $A,B\in\mathcal S$ such that $A\ne B$.

Clearly this property is invariant under bijections.

If we take $A=\mathbb Q$ and we take for every real number $r$ a sequence $(q_n)_{n\in\mathbb N}$ of rationals, which converges to $r$, then the system of sets of the form $$A_r=\{q_n; n\in\mathbb N\}$$ for every $r\in\mathbb R$ give as an almost disjoint family of cardinality $|\mathbb R|=\mathfrak c$.

Such almost disjoint systems appeared in several other questions at MSE, for example here and here.

Answer 2

Suppose that $|A|=\aleph_0$. We are asking whether there exists a chain in $(\mathcal P(A),\subseteq)$ that has cardinality $\mathfrak c$.

Clearly this property is invariant under bijections.

If we take $A=\mathbb Q$ and $$A_r=(r,\infty)\cap\mathbb Q$$ for any real number $r\in\mathbb R$, then $\{A_r; r\in\mathbb R\}$ is a chain of cardinality $|\mathbb R|=\mathfrak c$.

This proof have appeared in an answer here.