Local connectedness “with respect to”. (English) Zbl 1006.54025
Summary: A certain new concept of connectedness in frames is introduced, namely, “local connectedness with respect to”. We show that whenever \(h:L\to M\) is a dense homomorphism with \(M\) locally connected with respect to \(h\), then \(h\) preserves connectedness. (This provides a “partial” converse to a result of D. Baboolal and B. Banaschewski [J. Pure Appl. Algebra 70, No. 1/2, 3-16 (1991; Zbl 0722.54031)].) Also, under the hypothesis, the right adjoint preserves pairwise disjoint joins.
MSC:
54D05 | Connected and locally connected spaces (general aspects) |
06D22 | Frames, locales |
54E15 | Uniform structures and generalizations |