Some types of relative paracompactness

Vladimir Pavlovic
2003 Mathematica Moravica  
This paper is a continuation of the study of relative topological properties. We use a characterization of paracompactness via a certain selection principle to introduce five types of relative paracompactness, provide examples showing that none of them coincide with each other and establish some results concerning finite unions of subspaces which are relatively paracompact in one or another of the defined senses. Proof. Let a sequence U k,i : k, i < ∞ be given such that U k,i ⊆ T X and such
more » ... any of the families U k,i covers X. Apply the fact that 3 − (F 1 |X) lf to each of the sequences U k,i : i < ∞ in order to obtain for each k ∈ N a sequence L k,i : i < ∞ such that L k,i ⊆ T X , families L k,i are all locally finite on F 1 , each of the sets i<∞ L k,i covers X and such that L k,i U k,i . Now apply 3 − (F 2 |X) lf to the sequence i<∞ L k,i : k < ∞ so as to obtain a sequence J k : k < ∞ , where J k ⊆ T X , each J k is locally finite on F 2 , the family k J k covers X and J k i<∞ L k,i . For each k, i put J k,i = {A ∈ J k : ∃U ∈ L k,i (A ⊆ U )} and take any g k,i : and as L k,i is locally finite on F 1 , we conclude that J k,i must also be locally finite on F 1 . The family J k,i is locally finite on F 2 (because such is J k ), so (in view of Lemma 1. as well as the way in which we constructed the J k,i -s) the family J k,i is also locally finite on F 2 . And so, we have defined the families J k,i ≺ U k,i of open sets each of which is locally finite on F 1 ∪ F 2 . Since J k i<∞ L k,i it must be J k = i<∞ J k,i , and therefore:
doi:10.5937/matmor0307033p fatcat:qqgzmsynujdijpvcvi6z6kgube