When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from <http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf>


I could not reproduce theorem 5.7:


Every solvable set (tilable) has a solution S such that every finite block occurring in S also occurs infinitely often in S:


Proof in the text: 
Given a solution T and the set K of all finite blocks occurring in T, consider the set L of all subsets of K such that a subset A of K belongs to the L if there is a solution covered by A, i.e., in that solution all occurring finite blocks belong A. The set L is not empty because K belongs to it and it has minimal members. ...


Here's my problem: I could not see why the minimal member exists. There might be a sequence of A1>=A2>=A3>= ... belonging to L but their intersection does not belong to it ... I lacked a proof here... Any idea?



An established proof in Wang Tile which I doubt
Note that the set of solutions of a set $A$ is a closed set (with respect to the product topology): every tiling that is not a solution of $A$ has an occurrence of a finite pattern $p$ not in $A$, and the cylinder set around $p$ (the set of all tilings with pattern $p$ on the same position) is an open set consisting only of tilings that are not solutions of $A$.


Now, if you have a chain of sets $A\_1\supseteq A\_2\supseteq\cdots$ all with non-empty solution sets, their solution sets form a decreasing chain $X\_1\supseteq X\_2\supseteq \cdots$ of closed sets. Since the space is compact, the intersection $X\_\infty:=\bigcap\_i X\_i$ is a non-empty closed set. Moreover, the intersection $X\_\infty$ is precisely the set of solutions of $A\_\infty:=\bigcap\_i A\_i$. Hence, $A\_\infty$ is indeed in the partially ordered set $\mathcal{L}$.


Now you can apply Zorn's lemma.


