engleski

Torus-like continua which are not self-covering spaces

For each non-quadratic p-adic integer, p>2, we give an example of a torus-like continuum Y (i.e. inverse limit of an inverse sequence, where each term is the 2-torus T&sup2 ; and each bonding map is a surjective homomorphism), which admits three non-equivalent 4-sheeted coverings f₁ :X₁ → Y, f₂ :X₂ → Y, f₃ :X₃ → Y such that the total spaces X₁ =Y, X₂ and X₃ are pair-wise non-homeomorphic. Furthermore, Y admits a p-sheeted covering f₄ :X₄ → Y, although each bonding map of Y is a p-sheeted covering of T&sup2 ; . In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.

Inverse system; direct system; h-connected space; covering mapping; torus-like continuum; p-adic number; quadratic number

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