## $$R_z$$-supercontinuous functions.(English)Zbl 1349.54043

Commencing with the introduction of a new concept ‘$$r_z$$-open’, the concept of $$R_z$$-supercontinuity is brought in. Other variants of continuity are compared to this new concept; the non-reversibility of the implications is justified by suitable examples.
As usual, $$R_z$$-supercontinuity is explained by giving some equivalent conditions; $$R_z$$-convergence along with its ‘iff’ condition is given.
The resemblance of $$R_z$$-supercontinuity with continuity is delineated in the usual way.
The concept of $$R_z$$-homeomorphism is introduced in the usual manner. It is shown that if $$X$$ is an $$r_z$$-completely regular space and $$f\:X\rightarrow Y$$ is an $$R_z$$-homeomorphism, then $$X$$ and $$Y$$ are homeomorphic completely regular spaces.
The discussion is continued with the introduction of $$r_z$$-closed graph and its natural properties.
The newly introduced concepts like $$r_z$$-quotient topology, $$r_{cl}$$-quotient topology, $$z$$-quotient topology and $$r$$-quotient topology are duly compared; a new topology $$\tau_{rz}$$ on $$X$$ is defined by the collection of all $$r_z$$-open sets and it is shown that the $$r_z$$-quotient topology on $$Y$$ determined by the function $$f\:(X,\tau)\rightarrow Y$$ is identical with the standard quotient topology on $$Y$$ determined by $$f\:(X,\tau_{rz})\rightarrow Y$$.

### MSC:

 54C08 Weak and generalized continuity 54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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