On Iso-dense and Scattered Spaces without $$\textbf{AC}$$
Abstrak
AbstractA topological space is iso-dense if it has a dense set of isolated points, and it is scattered if each of its non-empty subspaces has an isolated point. In $$\textbf{ZF}$$ ZF (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice ($$\textbf{AC}$$ AC )), basic properties of iso-dense spaces are investigated. A new permutation model is constructed, in which there exists a discrete weakly Dedekind-finite space having the Cantor set as a remainder; the result is transferable to $$\textbf{ZF}$$ ZF . This settles an open problem posed by Keremedis, Tachtsis and Wajch in 2021. A metrization theorem for a class of quasi-metric spaces is deduced. The statement “Every compact scattered metrizable space is separable” and several other statements about metric iso-dense spaces are shown to be equivalent to the axiom of countable choice for families of finite sets. Results related to the open problem of the set-theoretic strength of the statement “Every non-discrete compact metrizable space contains an infinite compact scattered subspace” are also included.
Penulis (3)
Kyriakos Keremedis
Eleftherios Tachtsis
Eliza Wajch
Akses Cepat
- Tahun Terbit
- 2023
- Bahasa
- en
- Total Sitasi
- 2×
- Sumber Database
- CrossRef
- DOI
- 10.1007/s00025-023-01926-2
- Akses
- Open Access ✓