I'm reading the book on [Injective choice functions by Holz, Podewski and Steffens](http://www.springer.com/gp/book/9783540172215), and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few situations I wasn't able to come up with examples myself (not due to lack of trying, but rather lack of skill). So that's what this thread is about.


A *covering* of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal U}\to X$ is called a *choice function* if $f(A)\in A$ for all $A\in {\cal U}$. A *marriage* is an injective choice function.


We say that ${\cal U}$ is *critial* if


1. there is a marriage $f:{\cal U}\to X$, and
2. every marriage $f:{\cal U}\to X$ is surjective.


I would like to see examples of the following:


1. Is there a critical covering on an infinite set $X$ that contains no singletons, and at least one member of the covering is infinite? (Noah Schweber demonstrated [here](https://mathoverflow.net/a/243602/8628) that at least one member of any critical covering has to be finite.)
2. Is there a critical covering ${\cal U}$ on $\omega$ such that $$\bigcup \{F\in {\cal U}: F\text { is finite}\} \neq \omega$$?


**Edit** I added label graph-theory because marriages in this context are matchings in a bipartite graph.



On some examples of critical families
For any fixed $n\in\mathbb N,$ there is a critical covering $\mathcal U$ of $\omega$ such that each finite member of $\mathcal U$ contains exactly $n$ elements, and one member of $\mathcal U$ is infinite; moreover, $\bigcup\{F\in\mathcal U:F\text{ is finite}\}\ne\omega.$


Namely, let
$$\omega=A\cup B\_1\cup B\_2\cup B\_3\cup\cdots$$
where $A,B\_1,B\_2,B\_3,\dots$ are pairwise disjoint sets, $|A|=1,$ and $|B\_i|=n+1$ for each $i,$ and let
$$\mathcal U=\{\omega\}\cup\binom{B\_1}n\cup\binom{B\_2}n\cup\binom{B\_3}n\cup\cdots.$$


