Near homeomorphisms and \(G\)-like continua (Q1368900)

Let \(G\) be a finite connected planar graph. Denote by \(C(G)\) the space of continuous and surjective self-maps of \(G\) endowed with the \(C^0\)-topology, and by \(\eta (G)\) its subset consisting of maps which cannot be uniformly approximated by embeddings of \(G\) into the plane. The main result asserts that \(\eta (G)\) is open and dense in \(C(G)\) provided \(G\) has a branch point. There is also given an example of a map \(f\in \eta (T)\), where \(T\) is a triod, for which the inverse limit \((T,f)\) can be embedded as the attractor of some planar homeomorphism.