A \(\Pi^0_4\)-hard space

Let \(X\) be the space made of three rectangles attached along a common edge.


In the article [1], we show that \(X\) is \(\Pi^0_4\)-hard, in the sense that in the Polish space of compact subsets of \(\mathbb{R}^3\) endowed with the Vietoris topology (equivalently, the Hausdorff metric), the collection \(\mathcal{H}_X\) of compact sets that are homeomorphic to \(X\) is \(\Pi^0_4\)-hard. The picture below illustrates the reduction of a \(\Pi^0_4\)-complete problem to \(\mathcal{H}_X\).
Given a sequence of infinite binary-valued matrices \(M^n_{i,j}\), one builds a subset of \(\mathbb{R}^3\) which is homeomorphic to \(X\) if and only if for every \(n\), almost every row of the matrix \(M^n\) contains at least one occurrence of \(1\).

[1] Guilhem Gamard, Mathieu Hoyrup and Alexis Terrassin. Borel complexity of continua. In preparation.