Every locally connected functionally Hausdorff space is c-resolvable

Abstract

For a cardinal κ, we say that a T1-space Y is a κ-retodic space if Y is partitioned by a family {Dξ}ξ∈κ of dense subsets such that the complement of every Dξ is totally disconnected. It is shown that “A locally connected space X is κ-resolvable if for every connected open subset U⊆X, there exists a κ-retodic space YU and a non-constant continuous function f:U⟶YU”. Consequently, every locally connected functionally Hausdorff space is c-resolvable, which is an answer to the question of K. Padmavally about resolvability of locally connected spaces, and also every c-irresolvable locally connected T1 space contains an open connected subset in which all continuous functions on it to any c-retodic space (all real continuous functions on it) are constant.