SOME WEAKER FORMS OF THE CHAIN (F) CONDITION FOR METACOMPACTNESS

Abstract

We define, in a slightly unusual way, the rank of a partially ordered set. Then we prove that if X is a topological space and W = {W(x) : x ∈ X } satisfies condition (F) and, for every x ∈ X , W(x) is of the form i∈n(x) W i (x), where W 0 (x) is Noetherian of finite rank, and every other W i (x) is a chain (with respect to inclusion) of neighbourhoods of x, then X is metacompact. We also obtain a cardinal extension of the above. In addition, we give a new proof of the theorem 'if the space X has a base B of point-finite rank, then X is metacompact', which was proved by Gruenhage and Nyikos.

Paracompactness of paratopological groups which are GO-spaces

“…In this paper, we show that any paratopological group which is a GO-space such that the multiplication in the group preserves the order, has a W satisfying condition (F ) as in the hypotheses of Corollary 2.4 in [4], and therefore it is paracompact.…”

“…Then we have [r −1 y, r y] ∈ W(y) and x ∈ [r −1 y, r y] ⊆ O wherexy −1 , y < x, yqy −1 , y = x, yx −1 , y > x.Collectionwise normality of GO-spaces, Corollary 2.4 in[4] and Theorem 2.2 give us the following.Corollary 2.3.Every order preserving GO-paratopological group G is paracompact.…”

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

“…In this work, the rank of a partially ordered set P, rank(P), is the smallest cardinal number such that for each subset A of P with |A| ≥ , A is dependent. In [9], using the concepts of rank and Noetherianness, some results related to metacompactness of the spaces have been obtained.…”