In 1983, M, Ito and K, Tamano interduced the notion of almost locally finiteness and ivestigated some properties of the class of spaces with a σ atmost locally finite base, In this paper, some more results about such spaces are given, The main results are as follows; (1) The image of a space with a σ-almost locally finite base under the finite to one closed map has a σ-almost locally finited base; (2) The locally finite sum theorem holds for such spaces; (3) A space X has a σ-almost locally finite base if and only if X is a paracompact σ-space and every closed subset of X has an almost locally finite open neighborhood base; (4) The monotonically normal space with an M structure has a a σ-almost locally finite base.

本文证明了闭遗传强1-星紧性与可数紧性等价；在正则空间中，闭遗传ω-星紧性与可数紧性等价。

It is an old problem posed by Bennett and Lutzer whether every perfect GO-space has a perfect orderable extension. As an approach to this problem, we prove that, for a perfect GO-space X, X has a perfect linearly ordered extension if and only if there is a o-discrete subset F such that GOx (ф, X-F, F, ф) is perfect, where GOx (ф, X-F, F, ф) is the ordered set X with the topology defined so that every point in F is isolated and every point in X-F has the usual interval neighborhood base.

A collection D of subsets of a space is minimal if each element of D contains a point which is not contained in any other element of D. A base of a topological space is σ-minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space X satisfying that X is not metrizable and every subspace of X has a σ-minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

In this paper we prove that if the underlying LOTS of a perfect GO-space satisfies local perfectness, then the GO-space can embed in a perfect LOTS.