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Between local connectedness and sum connectedness

Kohli J. K., Singh D., Tyagi B. K.

Commentationes Mathematicae

2013

Vol 53, No. 1

3--16

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 ( 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.

Quasi cl-supercontinuous functions and their function spaces

Kohli J. K., Aggarwal J.

Demonstratio Mathematica

2012

Vol. 45, nr 3

677-697

A new class of functions called 'quasi cl-supercontinuous functions' is introduced. Basic properties of quasi cl-supercontinuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the mathematical literature is elaborated. The notion of quasi cl-supercontinuity, in general, is independent of continuity but coincides with cl-supercontinuity (clopen continuity) (Applied General Topology 8(2) (2007), 293-300; Indian J. Pure Appl. Math. 14(6) (1983), 767-772), a significantly strong form of continuity, if range is a regular space. The class of quasi cl-supercontinuous functions properly contains each of the classes of (i) quasi perfectly continuous functions and (ii) almost cl-supercontinuous functions; and is strictly contained in the class of quasi z-supercontinuous functions. Moreover, it is shown that if X is sum connected (e.g. connected or locally connected) and Y is Hausdorff, then the function space Lq(X, Y ) of all quasi cl-supercontinuous functions as well as the function space Lq(X, Y ) of all almost cl-supercontinuous functions from X to Y is closed in Y X in the topology of pointwise convergence.