Connectedness and local cut points of generalized Sierpiński carpets

We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpiński carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpiński carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. On the other hand, we obtain two criteria: (1) for a connected GSC to have cut points, (2) for a connected GSC with no cut points to have local cut points. With these two criteria, we characterize all GSCs that are homeomorphic to the standard Sierpiński carpet.

Our results on cut points and local cut points hold for Barański carpets, too. Moreover, we extend the connectedness result to Barański sponges. Thus, we also characterize when a Barański carpet is homeomorphic to the standard GSC.

Keywords

generalized Sierpiński carpets, cut points, local cut points, connectedness, Hata graphs

2010 Mathematics Subject Classification

Primary 28A80. Secondary 54A05.

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Dedicated to the memory of Professor Ka-Sing Lau

Received 2 September 2022

Accepted 13 June 2023

Published 10 July 2024