**Abstract**:

The main purpose of the present work is to introduce and study a new class of semi-open sets called -open sets. We use this set to study new topological concepts such as -continuous functions, almost and weakly -continuous functions, contra -continuous functions, -compact and -connected spaces. Eventually, new types of -separation axioms are also studied.

At the beginning several properties of -open sets are obtained. It is proved that if a topological space is semi- , then the family of semi-open sets is equal to the family of -open sets. Moreover, if a topological space is semi-regular, then every open set is -open set. It is shown that for any space, every singleton subset of is -open if and only if it is semi-regular.

For -continuous functions, almost and weakly -continuous functions several results and characterizations are obtained. It is proved that the restriction of -continuous to -open, semi-closed set is also -continuous. Moreover, it is shown that if the co-domain is hyperconnected, then for any function is almost -continuous. Furthermore, we determine the relation between almost -continuous and weakly -continuous functions as follows for a function and is extremely disconnected, then is almost -continuous if and only if it is weakly -continuous.

The notions of -compact spaces and -sets are studied. It is showed that -compactness is strong form of semi-compactness. Also, if is -compact, then it is quasi H-closed and hence it is mildy compact. We prove that a space is -compact if and only if every proper -closed subset of is -set. Regarding to -connected space, we prove that a topological space is -connected if and only if the -closure of every non-empty -open set is. Finally we define generalized forms of semi-separation axioms are called -separation axioms where several characterizations are given and the relations among them are found.

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