Covering properties in \sigma-products
by
Keiko Chiba
Department of Mathematics, Faculty of Science, Shizuoka

We assume that each space is a Hausdorff space having at least two points.

Let X = \Pi_{\alpha in A}X_\alpha be a Cartesian product of spaces {X_\alpha|\alpha in A} with x^* = (x^*_\alpha)_{\alpha in A} in X. For each x = (x_\alpha)_{\alpha in A} in X, let Q(x) = {\alpha in A:x_\alpha =/= x^*_\alpha}. The subspace \sigma = {x in X : Q(X) is finite } of X is called the \sigma-product of {X_\alpha|\alpha in A}. We call x^* in \sigma a base point of \sigma. For a finite subset F of A, \Pi_{\alpha in F}X_\alpha is called a finite subproduct of \sigma.

A.P.Kombarov[1973] proved that
(i) If every finite subproduct of \sigma is paracompact, then \sigma is paracompact.
(ii) If every finite subproduct of \sigma is Lindelöf, then \sigma is Lindelöf.

The following question arises naturally.

Question. Suppose every finite subproduct of \sigma satisfies a topological property P. Does \sigma satisfy P ?

1. The answer is ``no'' for ``normality''([3]).

2. The answer is ``yes'' for many covering properties (i.e., metacompactness, hereditarily metacompactness, weak \theta-refinability, weak \delta\theta-refinability, etc; (under the assumption of \sigma is subnormal) subparacompactness; (under the assumption of \sigma is normal) submetacompactness, submeta-Lindelöfness, etc.;[metacompactness and subparacompactness are due to H.Teng[4] and others are due to Chiba[1, 2]]).

3. The answer is ``no'' for strong paracompactness, orthocompactness and star-Lindelöfness([3]).

[1] K.Chiba, The submetacompactness of \sigma-products, Math. Japonica, 36, No.4(1991), 711-715.

[2] K.Chiba, Covering properties in \sigma-products, Math. Japonica, 39, No.2(1994), 343-352.

[3] K.Chiba, The strong paracompactness of \sigma-products, to appear in Math. Japonica, 51, No.3.

[4] H.Teng, On \sigma-product spaces I, Math. Japonica, 36, No.3(1991).

Date received: June 29, 1999