I'm reading the book on Injective choice functions by Holz, Podewski and Steffens, 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

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

I would like to see examples of the following:

- 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 that at least one member of any critical covering has to be finite.)
- 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.