\(R_z\)-supercontinuous functions.

*(English)*Zbl 1349.54043Commencing 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 \(f\:(X,\tau_{rz})\rightarrow Y\).

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 \(f\:(X,\tau_{rz})\rightarrow Y\).

Reviewer: M. N. Mukherjee (Calcutta)

##### MSC:

54C08 | Weak and generalized continuity |

54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |