The authors introduce and study a new family of theorems extending the class of Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form “the image of a subset that is large in some sense is a singleton” with requirements of the milder form “the image of a subset that is large in some sense is a subset that is small in some sense”. This approach covers the case of mappings \(S^m\rightarrow \mathbb{R}^n\) with \(m<n\) and extends to wider classes of spaces.
The main theorems are as follows:
Theorem 4: Let \(f\) be a continuous map of the boundary \(\partial\Delta^n\) of the \(n\)-dimensional complex \(\Delta^n\) to a contractible metric space \(M\). Then a set of spherical \(f\)-neighbors intersects all facets of \(\Delta^n\).
Theorem 6: Let \(C\) be a KKM cover of the \(n\)-sphere \(S^n\), and let \(f:\mathbb{S}^n\rightarrow M\) be a continuous map to a contractible metric space \(M\). Then a set of spherical \(f\)-neighbors intersects all elements of \(C\).
Theorem 19: Let \(X\) be a compact normal space, let \(M\) be a contractible metric space and let \(f:X\rightarrow M\) be a continuous map. Then, for any non-nullhomotopic cover \(C\) of \(X\), a set of spherical \(f\)-neighbors intersects at least \(rk(C)+1\) elements of \(C\). In particular, for any principal cover, a set of spherical \(f\)-neighbors intersects all elements of the cover.
Theorem 36: Let \(A\) be a compact normal space, Let \(C\) be a closed cover of \(A\), and let \([C]\) be the corresponding homotopy class in \([A,N(C)]\), where \(N(C)\) is the nerve. Let \(Z\) be a normal space containing \(A\) as a subspace. If the triple \((Z,A,[C])\) is Eilenberg-Pontryagin, with \(EP\) rank \(rk(Z,A,[C])>0\), then for any metric space \(M\) and any continuous map \(f:A\rightarrow M\) that extends to a continuous map \(Z\rightarrow M\), a set of spherical \(f\)-neighbors intersects at least \(rk(Z,A,[c])+1\) elements of \(C\).
The authors also give an open problem:
Question: Which of the other versions of the topological Helly theorem give sufficient conditions for principal and non-nullhomotopic covers?