Between local connectedness and sum connectedness

J.K. Kohli ; D. Singh ; B.K. Tyagi

Commentationes Mathematicae, Tome 53 (2013) no. 1, pp. 3-16.

Résumé

A new generalization of local connectedness called Z-local connectedness is introduced. Basic properties of Z-locally connected spaces are studied and their place in the hierarchy of variants of local connectedness, which already exist in the literature, is elaborated. The class of Z-locally connected spaces lies strictly between the classes of pseudo locally connected spaces (Commentations Math. 50(2)(2010),183-199) and sum connected spaces ($\equiv$ weakly locally connected spaces) (Math. Nachrichten 82(1978), 121-129; Ann. Acad. Sci. Fenn. AI Math. 3(1977), 185-205) and so contains all quasi locally connected spaces which in their turn contain all almost locally connected spaces introduced by Mancuso (J. Austral. Math. Soc. 31(1981), 421-428). Formulations of product and subspace theorems for Z-locally connected spaces are suggested. Their preservation under mappings and their interplay with mappings are discussed. Change of topology of a Z-locally connected space is considered so that it is simply a locally connected space in the coarser topology. It turns out that the full subcategory of Z-locally connected spaces provides another example of a mono-coreflective subcategory of TOP which properly contains all almost locally connected spaces.

Détail

DOI : 10.14708/cm.v53i1.768

Mots-clés : Z-locally connected space, almost (quasi, pseudo) locally connected space, sum connected space, regular open set, regular $F_\sigma$-set, $\theta$-open set, cl-supercontinuous function, mono-coreflective subcategory