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Given a complete simple topological graph $G$, a $k$-face generated by $G$ isthe open bounded region enclosed by the edges of a non-self-intersecting$k$-cycle in $G$. Interestingly, there are complete simple topological graphswith the property that every odd face it generates contains the origin. In thispaper, we show that every complete $n$-vertex simple topological graphgenerates at least $\Omega(n^{1/3})$ pairwise disjoint 4-faces. As an immediatecorollary, every complete simple topological graph on $n$ vertices drawn in theunit square generates a 4-face with area at most $O(n^{-1/3})$. Finally, weinvestigate a $\mathbb Z_2$ variant of Heilbronn triangle problem.

Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems