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On the homology of the Harmonic Archipelago

Umed Karimov, Dušan Repovš (2012)

Open Mathematics

We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.

On the hyperspace C n ( X ) / C n K ( X )

José G. Anaya, Enrique Castañeda-Alvarado, José A. Martínez-Cortez (2021)

Commentationes Mathematicae Universitatis Carolinae

Let X be a continuum and n a positive integer. Let C n ( X ) be the hyperspace of all nonempty closed subsets of X with at most n components, endowed with the Hausdorff metric. For K compact subset of X , define the hyperspace C n K ( X ) = { A C n ( X ) : K A } . In this paper, we consider the hyperspace C K n ( X ) = C n ( X ) / C n K ( X ) , which can be a tool to study the space C n ( X ) . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.

On the LC1-spaces which are Cantor or arcwise homogeneous

Hanna Patkowska (1993)

Fundamenta Mathematicae

A space X containing a Cantor set (an arc) is Cantor (arcwise) homogeneousiff for any two Cantor sets (arcs) A,B ⊂ X there is an autohomeomorphism h of X such that h(A)=B. It is proved that a continuum (an arcwise connected continuum) X such that either dim X=1 or X L C 1 is Cantor (arcwise) homogeneous iff X is a closed manifold of dimension at most 2.

On the metric dimension of converging sequences

Ladislav, Jr. Mišík, Tibor Žáčik (1993)

Commentationes Mathematicae Universitatis Carolinae

In the paper, some kind of independence between upper metric dimension and natural order of converging sequences is shown — for any sequence converging to zero there is a greater sequence with an arbitrary ( 1 ) upper dimension. On the other hand there is a relationship to summability of series — the set of elements of any positive summable series must have metric dimension less than or equal to 1 / 2 .

On the Separation Dimension of K ω

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that t r t K ω > ω + 1 , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

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