Fuzzy θ-semi-generalized closed sets

In this paper, a new notion of the fuzzy generalized closed sets called fuzzy θ-semi-generalized closed sets in fuzzy topological spaces is introduced and studied their properties. Furthermore, we study these new sets in relation to some other types of already known fuzzy sets.


Introduction.
The concept of fuzzy sets due to Zadeh [16] naturally plays important role in the study of fuzzy topological spaces which has been introduced by Chang [4]. In the year of 1970, Levine [7] initiated the study of generalized closed set in general topology. Generalized closed (open) sets play an important role in general topology. After that, Balasubramanian and Sundaram [3] defined fuzzy generalized closed set in fuzzy topological spaces. Later, El-Shafei [5] introduced fuzzy semigeneralized closed and fuzzy generalized semi-closed set with some of their properties in year 2007.
In this paper, we introduced another new notion of fuzzy generalized closed set called fuzzy θsemi-generalized closed set which is an alternative generalization of fuzzy semi-closed set by utilizing semi-θ-closure operator in fuzzy topological spaces. The relationships among some fuzzy generalized closed sets and this new notion are obtained. Some examples and their properties were discussed in details.

Preliminaries.
Throughout this paper, let X be a set and I the unit interval. A fuzzy set in X is an element of the set of all functions from X to I. The family of all fuzzy sets in X is denoted by X I . A fuzzy singleton x  is a fuzzy set in X define by   and hence fuzzy semi-θ-closed set is a fuzzy semi-closed set.

Fuzzy θ-Semi-Generalized Closed Sets.
In this section, we introduce fuzzy θ-semi-generalized closed sets in fuzzy topological space and study some of their characteristics and their relationships with other notions.
The complement of fuzzy θ-semi generalized closed set is fuzzy θ-semi generalized open set (briefly f-θsg-open set).
The family of all fuzzy semi-open sets is Hence the family of all fuzzy semi-closed sets is Lemma 3.2. Every fuzzy θ-semi-generalized closed set is fuzzy semi-generalized closed.

Lemma 3.3.
Every fuzzy θ-generalized closed set is fuzzy θ-semi-generalized closed but the converse may not be true in general. Base on the discussion above, every fuzzy semi-closed set is fuzzy semi-generalized closed set but the converse is not true (see [5]). Moreover, fuzzy semi-θ-closed implies fuzzy θ-semi-generalized closed but the converse may not be true as in Example 3.1. Lemma 3.1 shows that fuzzy θ-semigeneralized closed imply fuzzy semi-generalized closed but the reverse implication is not true in general as shown in Example 3.2. Example 3.3 above shows that fuzzy θ-semi-generalized closed set does not implies fuzzy θ-generalized closed. Furthermore, fuzzy θ-closed implies fuzzy θ-generalized closed but the converse is not true (see [6]). The Figure 1 below summarizes the relationships among some fuzzy generalized closed sets discussed above where none of these implications are reversible.

Some Properties of Fuzzy θ-Semi-Generalized Closed Sets.
In this section, we study some properties of fuzzy θ-semi-generalized closed set.
(iii) is the complement of (ii).
(iv) is the complement of (i).
(v) Observe that  Similar argument as previous discussion, we will have Proposition 4.1. The union of two fuzzy θ-semi-generalized closed sets is always fuzzy θ-semigeneralized closed set.
Proof. Suppose that  and  are fuzzy θ-semi-generalized closed sets in X and let Intersection of two fuzzy θ-semi-generalized closed sets is not necessarily a fuzzy θ-semi-generalized closed set as the following example.