New Structures of Continuous Functions

: Continuous functions are novel concepts in topology. Many topologists contributed to the theory of continuous functions in topology. The present authors continued the study on continuous functions by utilizing the concept of gp α -closed sets in topology and introduced the concepts of weakly, subweakly and almost continuous functions. Further, the properties of these functions are established.


Introduction:
Weak continuity due to Levine 1 is one of the most important weak forms of continuity in topological spaces. The notion of sub-weakly continuous functions is investigated in 1984 and the relationship between weak continuity and subweakly continuity is studied. After that some topologists has discovered additional characteristics relating to sub-weakly continuous functions. Noiri demonstrated in 4 that the graph of a function is closed if the range space of a weakly continuous function is Hausdroff. A function p:M→ N between any two topological spaces M and N is continuous only if it is also both weakly continuous and ω * -continuous, according to Levine. By substituting a strictly weaker condition known as locally weak ω *continuity for continuity, Levines decomposition of continuity 2 was strengthened.
In the year 2018, Patil 3,4 studied the concept and properties of gpα-closed sets. The study of gpα-closed sets continued by defining the properties of gpα-closure, gpα-interior, gpαlimit points, gpα-continuous functions and gpαhomeomorphisms. Also, continued in by studying the properties such as, gpα-separation axioms, gpα-regular and gpα-normal spaces 5 .

Definition. 4:
If each gp-open cover in a space M has a finite sub-cover, then the space is said to be a gp-compact.
Theorem. 19: Every a.gpα-c function is w.gpα-c, but not conversely. Theorem. 25: For each m 1 , m 2 ∈ M, ∃ a function p on M into a T 2 -space N such that p(m 1 )≠p(m 2 ), p is w.gpα-c at m 1 and p is a.gpα-c at m 2 , then M is gpα-T 2 .

Conclusion:
In this present work, analysed new weaker form of some types of continuous functions namely weakly gp-continuous functions and subweakly gp-continuous functions. Also established the properties and some preservation theorems of weakly gp-continuous functions and subweakly gp-continuous functions. Further, almost gpcontinuous functions are studied here. There is a scope to study and extend these newly defined concepts in topological spaces.