Quasi Locally Connected Spaces and Pseudo Locally Connected Spaces

Two new generalizations of locally connected spaces called ‘quasi locally connected spaces’ and ‘pseudo locally connected spaces’ are introduced and their basic properties are studied. The class of quasi locally connected spaces properly contains the class of almost locally connected spaces (J. Austral. Math. Soc. 31(1981), 421–428) and is strictly contained in the class of pseudo locally connected spaces which in its turn is properly contained in the class of sum connected spaces (Math. Nachrichten 82(1978), 121-129; Ann. Acad. Sci. Fenn. A I Math. 3(1977), 185–205). Product and subspace theorems for quasi (pseudo) locally connected spaces are discussed. Their preservation under mappings and their interplay with mappings are outlined. Function spaces of quasi (pseudo) locally connected spaces are considered. Change of topology of a quasi (pseudo) locally connected space is considered so that it is simply a locally connected space in the coarser topology. In contradistinction with almost locally connected spaces, quasi (pseudo) locally connected spaces constitute a coreflective subcategory of TOP.