Screening properties of the subbase of all closed connected subsets of a connectedly generated space

J. L. Hursch
1970 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. In [1] de Groot has introduced the notation "connectedly generated" (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6]. If X is eg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and N v is any neighbourhood of p, then the complement
more » ... en the complement of N p may be covered by a finite number of closed connected sets, none of which contain p. In [1] and in [2] the concept of "screening" is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2. In this paper we confine our attention to the subbase ^ of closed connected sets, and the base Se of finite unions of members of ^ in a eg space. Using the definitions in [2], we show, in § 3, that subbase-regularity, with respect to ^, is equivalent to local connectedness for regular, eg spaces. In § 4, we show that, for a eg space X, subbase-normality with respect to *$ is equivalent to "locally connected and normal with respect to closed, connected sets". Also we give two other equivalent screening properties, one basic and the other subbasic, and an example which shows that (sub)base-regularity is not equivalent to (sub)base-normality for { c é')S §, The closure of a set A is denoted by Â, and its boundary by B(A). The notation A\B stands for {x: x £ A and x (£ B). Screening. If s/ is any collection of subsets of a setX, and if B and C are any two subsets of X which are disjoint, then we say "$/ screens B and C" if there exists a finite collection of sets in se such that their union is X, and no member of the collection intersects both B and C. We let s/{B, C) stand for the smallest integer n such that there exist A\, . . . , A n £ s/ and
doi:10.4153/cjm-1970-075-0 fatcat:6wazi36jh5d5pkmqpw2lljnrvm