In this paper, we discuss the local properties of quasihyperbolic mappings in metric spaces, which are related to an open problem raised by Huang et al in 2016. Our result is a partial solution to this problem, which is also a generalization of the corresponding result obtained by Huang et al in 2016.

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Suppose that G E and G ′ E ′ are domains, where E and E ′ denote real Banach spaces with dimension at least 2, and f : G → G ′ is a homeomorphism. The aim of this paper is to prove the validity of the implications:and the invalidity of their opposite implications, i.e., f is locallygives a negative answer to one of the open problems raised by Väisälä in 1999.

In this paper, we investigate the removability of φ -natural domains in Banach spaces. Let E be a real Banach space with dimension at least 2, G a domain in E , and let be a countable subset of G which satisfies a quasi hyperbolic separation condition. Then, G is φ -natural if and only if G is ψ -natural, quantitatively.