Kenichi Tamano

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 Kodamas question whether or not every Σ-product of Lasnev 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 Lasnev 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.

Lindelöf powers and products of function spaces

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable σ-compact spaces. All spaces considered below are assumed to be Tychonoff (= completely regular Hausdorff). We denote by Cp(X) the space of all continuous real-valued functions endowed with the topology of pointwise convergence on X ; this topology can be obtained as the restriction of the Tychonoff product topology on the set R of all real-valued functions onX to its subset C(X) (see [Arh1]). Cp(X, 2) is the subspace of Cp(X) consisting of all functions to 2 = {0, 1}. The symbols ω, R, I and C stand for the set of naturals, the real line, the segment [0, 1], and the Cantor cube 2. If P and Q are sets, then P denotes the set of all functions from Q to P ; if κ is a cardinal, then X is the κth power of X (with the Tychonoff product topology); the projection of X to its ith factor is denoted by πi. For j ∈ 2 and σ ∈ 2, denote σaj = σ ∪ {〈i, j〉} ∈ 2. The symbol c denotes the cardinality of the continuum. Polish spaces are separable completely metrizable spaces.

Generalized paracompactness of subspaces in products of two ordinals

Abstract We will characterize metacompactness, subparacompactness and paracompactness of subspaces of products of two ordinal numbers. Using them we will show: 1. For such subspaces, weak submetaLindelofness, screenability and metacompactness are equivalent. 2. Metacompact subspaces of ω 1 2 are paracompact. 3. Metacompact subspaces of ω 2 2 are subparacompact. 4. There is a metacompact subspace of ( ω 1 +1) 2 which is not paracompact. 5. There is a metacompact subspace of ( ω 2 +1) 2 which is not subparacompact.

A cosmic space which is not a μ-space

Abstract We prove that there exists a zero-dimensional space with a countable network which is not a μ -space, giving a negative answer to Tamanos question and a partial negative answer to Nagamis question.

Perfect κ-Normality of Product Spaces

ABSTRACT. A space X is called perfectly K‐normal (respectively, Klebanov) if the closure of every open set (respectively, every union of zero‐sets) in X is a zero‐set. It is proved: The product of infinitely many Lašnev spaces need not be perfectly K‐normal, in particular, S(ω1)2χ Dω1 is not perfectly K‐normal; a locally compact, paracompact space Y is Klebanov if and only if XχY is perfectly K‐normal for every Lašnev space X; if XχY is perfectly K‐normal for every paracompact s̀‐space X, then Y is perfectly normal. Properties of a Klebanov space are also studied.