Sequential fans in topology

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( ω 2) × S( ω 2) × ω 1, 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( ω 1) is normal.