Compatible algebraic structures on scattered compacta
| Autor: | Taras Banakh, Oleg Gutik, M. Rajagopalan |
|---|---|
| Rok vydání: | 2005 |
| Předmět: |
Discrete mathematics
Scattered space Pure mathematics Mathematics::General Mathematics Discrete space Hausdorff space Semilattice Mathematics::General Topology Pospace Separable space Topological semilattice Topological inverse semigroup Inverse semigroup Mathematics::Logic Scattered height Compactification (mathematics) Geometry and Topology Remainder Subspace topology Mathematics |
| Zdroj: | Topology and its Applications. 153(5-6):710-723 |
| ISSN: | 0166-8641 |
| DOI: | 10.1016/j.topol.2005.01.004 |
| Popis: | It is proved that each hereditarily collectionwise Hausdorff compact scattered space with finite scattered height admits a continuous semilattice operation turning it into a topological semilattice with open principal filters. On the other hand a compactification γN of a countable discrete space N whose remainder is homeomorphic to [0,ω1] admits no (separately) continuous binary operation turning γN into an inverse semigroup (semilattice). Also we construct a compactification ψN of N admitting no separately continuous semilattice operation and such that the remainder ψN∖N is homeomorphic to the one-point compactification of an uncountable discrete space. To show that ψN admits no continuous semilattice operation we prove that the set of isolated points of a compact scattered topological semilattice X of scattered height 2 is sequentially dense in X. Also we prove that each separable scattered compactum with scattered height 2 is a subspace of a separable compact scattered topological semilattice with open principal filters and scattered height 2. This allows us to construct an example of a separable compact scattered topological semilattice with open principal filters and scattered height 2, which fails to be Fréchet–Urysohn. Also we construct an example of a Fréchet–Urysohn separable non-metrizable compact scattered topological semilattice with open principal filters and scattered height 2. |
| Databáze: | OpenAIRE |
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