Sequential fans in topology

Katsuya Eda, Gary Gruenhage, Piotr Koszmider, Kenichi Tamano, Stevo Todorčević

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

For an index set I, let S(I) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems. For example, consider the problem which ask us to compute the tightness of the square of S(I). We shall show that this is in fact equivalent to the well-known and more crucial topological question of W. Fleissner which asks whether, in the class of first countable spaces, the property of being collectionwise Hausdorff at certain levels implies the same property at higher levels. Next, we consider Kodama's question whether or not every Σ-product of Lašnev spaces is normal. The sequential fan again enters the scene as we show S(ω2) × S(ω2) × ω1, which can be embedded in a Σ-product of Lašnev spaces as a closed set, can be nonnormal in some model of set theory. On the other hand, we show that the Σ-product of arbitrarily many copies of the slightly smaller fan S(ω1) is normal.

Original languageEnglish
Pages (from-to)189-220
Number of pages32
JournalTopology and its Applications
Volume67
Issue number3
DOIs
Publication statusPublished - 1995 Dec 8
Externally publishedYes

Keywords

  • Collectionwise Hausdorffness
  • Lašnev space
  • Normality
  • Product
  • Sequential fan
  • Tightness
  • Σ-product

ASJC Scopus subject areas

  • Geometry and Topology

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  • Cite this

    Eda, K., Gruenhage, G., Koszmider, P., Tamano, K., & Todorčević, S. (1995). Sequential fans in topology. Topology and its Applications, 67(3), 189-220. https://doi.org/10.1016/0166-8641(95)00016-X