SEMI-PRECONTINUOUS FUNCTIONS AND PROPERTIES OF GENERALIZED SEMI-PRECLOSED SETS IN TOPOLOGICAL SPACES

Andrijevíc (1986) introduced the class of semi-preopen sets in topological spaces. Since then many authors including Andrijević have studied this class of sets by defining their neighborhoods, separation axioms and functions. The purpose of this paper is to provide the new characterizations of semi-preopen and semi-preclosed sets by defining the concepts of semi-precontinuous mappings, semi-preopen mappings, semi-preclosed mappings, semi-preirresolute mappings, pre-semipreopen mappings, and pre-semi-preclosed mappings and study their characterizations in topological spaces. Recently, Dontchev (1995) has defined the concepts of generalized semi-preclosed (gsp-closed) sets and generalized semi-preopen (gsp-open) sets in topology. More recently, Cueva (2000) has defined the concepts like approximately irresolute, approximately semi-closed, contra-irresolute, contra-semiclosed, and perfectly contra-irresolute mappings using semi-generalized closed (sg-closed) sets and semi-generalized open (sg-open) sets due to Bhattacharyya and Lahiri (1987) in topology. In this paper for gsp-closed (resp., gsp-open) sets, we also introduce and study the concepts of approximately semi-preirresolute (ap-sp-irresolute) mappings, approximately semi-preclosed (ap-semi-preclosed) mappings. Also, we introduce the notions like contra-semi-preirresolute, contra-semi-preclosed, and perfectly contra-semipreirresolute mappings to study the characterizations of semi-pre-T1/2 spaces defined by Dontchev (1995).

(iv) For each x ∈ X, the inverse of every neighborhood of f (x) is a semipreneighborhood of x.
(v) For each x ∈ X and each neighborhood N x of f (x), there is a semi- (iv)⇒(v). Let x ∈ X and N x be a neighborhood of f (x).
(iii) (vi). Suppose that (iii) holds and let A be a subset of X.
Conversely, suppose that (vi) holds for any subset A of X. Let F be a closed subset of Y . Then,   The proof is similar to Lemma 3.5.
Theorem 3.7. Let f : X → Y be a mapping. Then the following are equivalent: The following lemma is proved in [3].

Lemma 3.8. If U is open and A is semi-preopen then U ∩ A is a semi-preopen set.
Lemma 3.9. Let A be a semi-preopen set in a space X and suppose A ⊂ B ⊂ cl A, then B is a semi-preopen set.
Proof. Since A is a semi-preopen set in X, then there exists a preopen set U in X such that U ⊂ A ⊂ cl U. As A ⊂ B, U ⊂ A ⊂ B implies that U ⊂ B. Also, cl A ⊂ cl(cl U)= cl U , and thus, B ⊂ cl U . Hence U ⊂ B ⊂ cl U, which implies that B is a semi-preopen set.
Proof. Necessity: since A ∈ SPO(X), then there exists a preopen set U ⊂ X such that U ⊂ A ⊂ cl U . Let cl X and cl X 0 denote, respectively, the closure operator in X and The converse is easy and hence is omitted.

Lemma 3.12.
A is a semi-preopen set and A ≠ ∅. Then, int cl A ≠ ∅.
Proof. Let A be a semi-preopen set such that A ≠ ∅. Then by Lemma 3.11, clA = cl int cl A. If intclA = ∅ then cl A = ∅ implies A = ∅, which is in contradiction to the hypothesis. Hence, int cl A ≠ ∅.
We give the following definition.
Now, we prove the following theorem.
But, we know that arbitrary union of semi-preopen sets is a semi-preopen set, thus, we obtain that . This implies that f is a semi-precontinuous map. Lemma 3.16 (see [13]). Let {X α | α ∈ ∆} be a family of topological spaces and ΠA α a subset of ΠXα, where ΠX α denotes the product space. Then, Now, in view of Lemma 3.16, one can prove the following lemma.
The following theorem proved in [22] is the generalization of Lemma 3.17.
Theorem 3.19. Let X i and Y i be topological spaces and f i : X i → Y i be a semiprecontinuous mapping for i = 1, 2. Then a mapping f : , which is a semi-preopen set since arbitrary union of semi-preopen sets is a semi-preopen set. Hence by arguing as above, The following theorem is the generalization of Theorem 3.19, which can be proved in view of Theorems 3.18 and 3.19.

Semi-preopen functions.
We give the following definition.
Note that every open map is semi-preopen but not the converse, which is shown by the following example.
Then it is clear that τ and σ are topologies on X and Y , respectively. If Also, note that every preopen (resp., semi-open) map is a semi-preopen map.
We recall that a mapping f : X → Y is called semi-preopen (in the sense of Cammaroto and Noiri [8]) if f (U) ∈ SPO(Y ) for each U ∈ SO(X). Clearly, every semipreopen map (in the sense of Cammaroto and Noiri [8]) is a semi-preopen map as given by Definition 4.1.
We recall the following lemma.
Lemma 4.3 (see [8]). The following are equivalent for a subset A of a space X: Hence, f is a semi-preopen map.

Semi-preclosed functions.
We recall the following definition.
Note that every closed map is semi-preclosed but not the converse, which is shown by the following example. Clearly, f is a semi-preclosed map but it is not closed since f ({b}) = {b} which is not a σ -closed set. Hence the example.
Note that every preclosed map (resp., semi-closed map) is a semi-preclosed map. Thus, we state the following theorem.  (X, σ ). This shows that s-preclosed and preclosed maps are independent of each other.
Now, we can prove it for s-preclosed maps in the following theorem.

Semi-preirresolute functions
Definition 6.1. A function f : X → Y is called semi-preirresolute if the inverse image of each semi-preopen set in Y is a semi-preopen set in X.
Note that every semi-preirresolute map is semi-precontinuous but not the converse, which is shown by the following example. Then, clearly f is semi-precontinuous but it is not a semi-preirresolute map since f −1 ({m, n}) = {a, d} which is not a semi-preopen set in (Y , σ ).
Next, we characterize the semi-preirresolute mappings in the following theorem. (i) f is semi-preirresolute.

Proof. (i)⇒(ii). Assume x ∈ X and V is a semi-preopen set in Y containing f (x).
Since f is a semi-preirresolute and let W = f −1 (V ) be a semi-preopen set in X containing x and hence f ( (ii)⇒(iii). Assume that V ⊂ Y is a semi-preopen set containing f (x). Then by (ii), there exists a semi-preopen set G such that 1 (V )). This shows that cl(f −1 (V )) is a semi-preneighborhood of x.
(iii)⇒(i). Let V be a semi-preopen set in Y , then cl(f −1 (V )) is semi-preneighborhood of each x ∈ f −1 (V ). Thus, for each x is a semi-preinterior point of cl 1 (V )). Therefore, f −1 (V ) is a semipreopen set in X and hence f is a semi-preirresolute map.
We state the following theorems. Theorem 6.4. If f : X → Y is a preopen and preirresolute mapping, then f is a semi-preirresolute.
Recall the following theorem.  Proof. Let A ∈ SPO(Y ), then there exists a preopen set U ⊂ Y such that U ⊂ A ⊂ cl U . Then by Theorem 6.6, f −1 (cl U) = cl(f −1 (U )). Also, we have Since f is a preirresolute map, then f −1 (U ) is a preopen set in X, and hence f −1 (A) is a semi-preopen set in X. Thus, f is a semi-preirresolute map.
One can easily prove the following theorem.

Semi-prehomeomorphisms
Definition 7.1. A bijective mapping f : (X, τ) → (Y , σ ) from a space X into a space Y is called a semi-prehomeomorphism if both f and f -1 are semi-preirresolute mappings. Now, we characterize the semi-prehomeomorphism in the following theorem.
Theorem 7.2. Let f : (X, τ) → (Y , σ ) be a bijective mapping from a space X into a space Y . Then the following are equivalent: Proof. (i) (ii). Since f is a bijective map, both f and f −1 are semi-preirresolute functions. Definition 7.3. A property which is preserved under semi-prehomeomorphism is said to be a semi-pretopological property.

Pre-semipreopen functions.
In this section, we introduce the notion of presemipreopen mappings analogous to pre-semiopen mappings [10]. We prove the following theorem.
Theorem 9.6. Let f : X → Y and g : Y → Z be two maps such that g • f is a presemi-preclosed map. Then, (i) If f is a semi-preirresolute surjection, then g is a pre-semi-preclosed map. g is a semi-preirresolute injection, then f is a pre-semi-preclosed map. Proof. We prove (ii) only. Suppose A is an arbitrary semi-preclosed set in X. Since g • f is a pre-semi-preclosed map, then g • f (A) is a semi-preclosed set in Z. Since g is a semi-preirresolute injective map, we have g −1 (g • f (A)) = f (A), which is a semi-preclosed set in Y . This shows that f is pre-semi-preclosed.
Recall that a map f : X → Y is called M-preclosed [23] if the image of each preclosed set is a preclosed set.
Finally, we prove the following theorem.
Proof. Let f be a continuous M-preclosed injective map and A a semi-preclosed set in X. Then, there exists a preclosed set F in X such that int F ⊂ A ⊂ F and so Thus, f is a pre-semi-preclosed map.

Generalized semi-preclosed sets and their mappings.
We recall the following definition.
Definition 10.1 (see [15]). A subset A of a space X is called a generalized closed set (written as g-closed) set if cl A ⊆ U whenever A ⊆ U and U is open.
Clearly, every closed set is a g-closed set. The complement of a g-closed set in X is called generalized open, that is, g-open [15] set. So, every open set is a g-open set.  Clearly, every semi-preclosed set is a gsp-closed set. The complement of a gspclosed set is called generalized semi-preopen [11] (written as gsp-open). Every semipreopen set is a gsp-open set. The family of all gsp-closed (resp., gsp-open) sets of X is denoted by GSPF(X) (resp., GSPO(X)).
Note that every semi-preirresolute (i.e., sp-irresolute) map is ap-sp-irresolute but not the converse, which is shown by the following theorem. σ ) and F is a gsp-closed subset of (X, τ). Therefore, we have (F s ) * ⊆ ((f −1 (U )) s ) * = f -1 (U). This implies that f is an ap-spirresolute.
Proof. Suppose that f is an ap-sp-irresolute. Let A be an arbitrary subset of X such that A ⊆ U where U ∈ SPO(X). Then by hypothesis (A s ) * ⊆ (U s ) * = U . Thus, all subsets of X are gsp-closed and hence all are gsp-open sets. Therefore, for Converse follows from Theorem 10.5. Note that every pre-semi-preclosed map is an ap-sp-closed map but not the converse, which is shown by the following theorem.
for every semi-preclosed subset B of (X, τ). A, where B is a semi-preclosed subset of (X, τ) and A is gspopen subset of (Y , σ ). Therefore, we have ((f (B)) s ) * ⊆ (A s ) * . Then f (B) ⊆ (A s ) * which implies that f is an ap-semi-preclosed map.

Proof. Let f (B) ⊆
The easy proof of the following theorem is omitted. Theorem 10.9. Let f : (X, τ) → (Y , σ ) be a map from a topological space (X, τ) into a topological space (Y , σ ). If the semi-preopen and semi-preclosed subsets of (Y , σ ) coincide, then f is ap-semi-preclosed if and only if f (B) ∈ SPO(Y , σ ), for every semipreclosed subset B of (X, τ).
Also, we give the following definition. Note that every semi-preirresolute (i.e., sp-irresolute) map is ap-sp-irresolute but not the converse, which is shown by the following definition.
Note that every pre-semi-preclosed map is an ap-sp-closed map but not the converse, which is shown by the following theorem. (i) f is contra-sp-irresolute. (ii) The inverse image of each semi-preclosed set in Y is a semi-preopen set in X.
Next, we give the following definition.
Definition 10.13. A subset of a space X is called semi-proclopen if it is both a semi-preopen and a semi-preclosed set.
Definition 10.14. A function f : (X, τ) → (Y , σ ) is called perfectly contrasemi-preirresolute (written as perfectly contra-sp-irresolute) if the inverse of every semi-preopen set in Y is a semi-preclopen set in X.
We recall the following definition.
Next, we characterize the semi-pre-T 1/2 spaces by using the ap-sp-irresolute and ap-semi-preclosed mappings.