×

Embeddings into the product of generalized Sierpiński curves admitting extensions of maps. (English) Zbl 1331.54036

This paper is a continuation of the author’s earlier papers on embeddings into the Cartesian product of generalized Sierpiński curves [Houston J. Math. 39, No. 2, 717–732 (2013; Zbl 1276.54012)] (see U. Milutinović’s paper [Glas. Mat., III. Ser. 27, No. 2, 343–364 (1992; Zbl 0797.54045)] for details on generalized Sierpiński curves). The main theorem states: for a metric space \(X\) of weight \(\tau \geq \aleph_0\) and a family of mappings of \(X\) into itself \(\{f_i\}_{i\in{\mathbb N}}\), the set of all embeddings \(h: X\to\Sigma(\tau)^\omega\) such that \(\dim \overline{h(X)} \leq \dim X\) and each mapping \(hf_i h^{-1}\) has an extension \(f_i^\ast: \overline{h(X)} \to \overline{h(X)}\) is residual in \(C(X, \Sigma(\tau)^\omega)\).

MSC:

54F45 Dimension theory in general topology
54C25 Embedding
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
PDFBibTeX XMLCite