I'm trying to think of examples of intersecting families with infinite cardinality. (Recall an intersecting family, $\mathcal{A}$, is a collection of sets such that $\forall A,B \in \mathcal{A}, A\cap B \neq \emptyset$.)
I've come up with $\mathcal{A}=\{A \in 2^{\mathbb{N}}: \{1\}\in A\}$, which you could do for any singleton. I'm having trouble cooking up examples that are not of this form.
If there was a maximum $n\in \mathbb{N}$ I could make, $\mathcal{A}=\{A \in 2^{[n]}: |A|>\frac{n}{2}\}$, except the natural numbers are unbounded to certainly that does not work either.