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We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...
This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.
We show that a metric space X admits no sensitive commutative group action if it satisfies the following two conditions: (1) X has property S, that is, for each ε > 0 there exists a cover of X which consists of finitely many connected sets with diameter less than ε; (2) X contains a free n-network, that is, there exists a nonempty open set W in X having no isolated point and n ∈ ℕ such that, for any nonempty open set U ⊂ W, there is a nonempty connected open set V ⊂ U such that the boundary ...
We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy.
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