1 Introduction

In 1977, Husain [3] has initiated the concept of superharmonic-open sets, which is considered as a wider class of some known types of near-open sets. In 1983, Mashhour et al. [4, 5] defined the concept of S-continuity, but for a single-valued function \(f:(X,\tau)\rightarrow (Y,\sigma)\). Many topological properties of the above mentioned concepts and others have been established in [6, 7]. The purpose of this paper is to present the upper (resp. lower) superharmonic-continuous multifunction as a generalization of each of upper (resp. lower) super-continuous superharmonic multifunction in the sense of Berge [7] the upper (resp. lower) sub-continuous and the upper (resp. lower) precontinuous superharmonic multifunction due to Popa [1, 8] and also upper (resp. lower) α-continuous and upper (resp. lower) β-continuous superharmonic multifunctions as given in [9, 10] recently. Moreover, these new superharmonic multifunctions are characterized and many of their properties have also been established.

2 Preliminaries

The topological space or simply space which is used here will be given by \((X, \tau)\) and \((Y, \sigma)\). \(\operatorname{\tau-cl}(W)\) and \(\operatorname{\tau-int}(W)\) denote the closure and the interior of any subset W of X with respect to a topology τ. In \((X, \tau)\), the class \(\tau^{*}\subseteq P(X)\) is called a superharmonic topology on X if \(X \in\tau^{*}\) and \(\tau^{*}\) is closed under arbitrary union [3], \((X, \tau^{*})\) is a superharmonic-topological space or simply superharmonic space, each member of τ is superharmonic-open and its complement is superharmonic-closed [5], In \((X, \tau^{*})\), the superharmonic-closure, the superharmonic-interior and superharmonic-frontier of any \(A\subseteq X\) will be denoted by \(\operatorname{superharmonic-cl}(A)\), \(\operatorname{superharmonic-int}(A)\) and superharmonic-\(\operatorname{fr}(A)\), respectively, which are defined in [5] and likewise the corresponding ordinary ones. Meanwhile, for any \(x \in X\), we define

$$\tau^{*}(x)=\bigl\{ W\subseteq X: W\in\tau^{*}, x \in W\bigr\} . $$

In \((X,\tau)\), \(A \subseteq X\) is called super-open [11] if there exists \(U\in\tau\) such that \(U\subseteq A \subseteq\operatorname{\tau-cl}(U)\), while A is preopen [5] if \(A\subseteq \operatorname{\tau-int}(\operatorname{\tau-cl}(A))\). The families of all super-open and preopen sets in \((X,\tau)\) are denoted by \(SO(X,\tau)\) and \(PO(X,\tau)\), respectively. Moreover,

$$\tau^{\alpha}=SO(X,\tau)\cap PO(X,\tau) $$

and

$$\beta O(X,\tau)\supset SO(X,\tau) \cup PO(X,\tau). $$

\(A\in \tau^{\alpha}\) and \(A \in\beta O(X,\tau)\) are called a superharmonic-α-set [2] and a superharmonic-β-open set [6], respectively. A single-valued superharmonic multifunction \(f:(X,\tau)\rightarrow (Y,\sigma)\) is called superharmonic-S-continuous [5], if the inverse image of each open set in \((Y,\sigma)\) is \(\tau^{*}\)-supra open in \((X,\tau)\). For a superharmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\), the upper and the lower inverses of any \(B\subseteq Y\) are given by

$$F^{+} (B)=\bigl\{ x\in X:F(x)\subseteq B\bigr\} $$

and

$$F^{-}(B)=\bigl\{ x \in X:F(X)\cap B \ne\phi\bigr\} , $$

respectively. Moreover, \(F:(X,\tau)\rightarrow(Y,\sigma)\) is called upper (resp. lower) super-continuous [7], if for each \(V\in\sigma\), \(F^{+} (V)\in \tau\) (resp. \(F^{-} (V)\in\tau\)). If τ in super-continuity is replaced by \(SO(X,\tau)\), \(\tau^{\alpha},PO(X,\tau)\) and \(\beta O(X,\tau)\), then F is upper (resp. lower) sub-continuous [8], upper (resp. lower) superharmonic α-continuous [1], upper (resp. lower) precontinuous [9] and upper (resp. lower) superharmonic-β-continuous [10], respectively. A space \((X,\tau)\) is called superharmonic-compact [12], if every supraopen cover of X admits a finite subcover.

3 Supra-continuous superharmonic multifunctions

Definition 3.1

A superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\) is said to be:

(a) upper superharmonic-continuous at a point \(x\in X \) if for each open set V containing \(F(x)\), there exists \(W \in\tau^{*}(x)\) such that

$$F(W)\subseteq V; $$

(b) lower superharmonic-continuous at a point \(x \in X\) if for each open set V containing \(F(x)\), there exists \(W \in\tau^{*}(x)\) such that

$$F(W)\cap V \neq\phi; $$

(c) upper (resp. lower) superharmonic-continuous if F has this property at every point of X.

Any single-valued superharmonic function \(f:(X, \tau)\rightarrow (Y,\sigma)\) can be considered as a multi-valued one which assigns to any \(x \in X\) the singleton \(\{f(x)\}\). We apply the above definitions of both upper and lower superharmonic-continuous multifunctions to the single-valued case. It is clear that they coincide with the notion of S-continuous due to Mashhour et al. [5]. One characterization of the above superharmonic multifunction is established throughout the following result, of which the proof is straightforward, so it is omitted.

Remark 3.1

For a superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\), many properties of upper (resp. lower) semicontinuity [7] (resp. upper (lower)) F-continuity [9], upper (resp. lower) sub-continuity [1], upper (resp. lower) precontinuity [10] and upper (resp. lower) (G-continuity [10]) can be deduced from the upper (resp. lower) superharmonic-continuity by considering \(\tau^{*}= \tau\) (resp. \(\tau^{*} = \tau^{\alpha}\), \(\tau^{*}= SO(X,\tau)\), \(\tau^{*}= PO(X,\tau)\) and \(\tau^{*}= \beta O(X,\tau)\)).

Proposition 3.1

A superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\) is upper (resp. lower) superharmonic-continuous at a point \(x\in X\) if and only if for \(V \in\sigma\) with \(F(x)\subseteq V \) (resp. \(F(x)\cap V\neq\phi\)). Then \(x\in \operatorname{superharmonic-int}(F^{+} (V))\) (resp. \(x \in \operatorname{superharmonic-int}(F^{-} (V))\).

Lemma 3.1

For any \(A \in(X,\tau)\), we have

$$\operatorname{\tau-int}(A)\subseteq \operatorname{superharmonic-int}(A)\subseteq A \subseteq\operatorname{superharmonic-cl}(A) \subseteq\operatorname{\tau-cl}(A). $$

Theorem 3.1

The following are equivalent for a superharmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\):

(i) F is upper superharmonic-continuous;

(ii) for each \(x\in X\) and each \(V \in \sigma(F(x))\), we have \(F^{+} (V)\in\tau^{*}(x)\);

(iii) for each \(x \in X\) and each \(V \in \sigma(F(x))\), there exists \(W\in\tau^{*}\) such that

$$F(W)\subseteq V; $$

(iv) \(F^{+} (V)\in\tau^{*}\) for every \(V\in\sigma\);

(v) \(F^{-} (K)\) is superharmonic-closed for every closed set \(K\subseteq Y\);

(vi) \(\operatorname{superharmonic-cl}(F^{-} (B))\subseteq F^{-} (\operatorname{\tau-cl}(B))\) for every \(B\subseteq Y\);

(vii) \(F^{+}(\operatorname{\tau-int}(B))\subseteq \operatorname{superharmonic-int}(F^{+} (B))\) for every \(B\subseteq Y\);

(viii) \(\operatorname{superharmonic-fr}(F^{-}(B))\subseteq F^{-}(\operatorname{fr}(B))\) for every \(B\subseteq Y\);

(ix) \(F:(X, \tau^{*}) \rightarrow (Y,\sigma)\) is upper superharmonic-continuous.

Proof

(i) ⇔ (ii) and (i) ⇒ (iv): Follow from Proposition 3.1.

(ii) ⇔ (iii): This is obvious, since the arbitrary union of superharmonic-open set is superharmonic-open.

(iv) = (v): Let K be closed in Y, the result satisfies

$$F^{+}(Y\backslash K)=X\backslash F^{-}(K). $$

(v) ⇒ (vi): By putting \(K = \operatorname{\sigma-cl}(B)\) and applying Lemma 3.1.

(vi) ⇒ (vii): Let \(B\Rightarrow Y\), then \(\operatorname{\sigma-int}(B) \in \sigma\) and so \(Y \backslash\operatorname{\sigma-int}(B)\) is super-closed in \((Y,\sigma)\). Therefore by (vi) we get

$$X\backslash\operatorname{super-int}\bigl(F^{+} (B)\bigr)=\operatorname{super-cl}\bigl(X\backslash F^{+} (B)\bigr)\subseteq \operatorname{sub-cl}(X\backslash F^{+} \bigl(\operatorname{\sigma-int}(B)\bigr) $$

and

$$\operatorname{supra-cl}(F^{-}\bigl(Y \operatorname{\sigma-int}(B)\bigr) \subseteq F-\bigl(Y\backslash\operatorname{\sigma-int}(B)\bigr)\subseteq X \backslash F^{+} \bigl(\operatorname{\sigma-int}(B)\bigr). $$

This implies that

$$F^{+} \bigl(\operatorname{\sigma-int}(B)\bigr) \subseteq \operatorname{supra-int}\bigl(F^{+}(B)\bigr). $$

(vii) ⇒ (ii): Let \(x\in X\) be arbitrary and each \(V\in \sigma(F(x))\) then

$$F^{+} (V) \subseteq \operatorname{supra-int}\bigl(F^{+} (V)\bigr). $$

Hence \(F^{+} (V) \in\tau^{*}(x)\).

(viii) ⇔ (v): Clearly, a suprafrontier and frontier of any set is superharmonic-closed and closed, respectively.

(ix) ⇔ (iv): Follows immediately. □

Theorem 3.2

For a superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\), the following statements are equivalent:

(i) F is lower superharmonic-continuous;

(ii) for each \(X\in X\) and each \(V \in\sigma\) such that

$$F(x)\cap V \neq\phi\quad\textit{and}\quad F^{-} (V) \in \tau^{*}(x); $$

(iii) for each \(x\in X\) and each \(V \in\sigma\) with \(F(x)\cap V \neq \phi\), there exists \(W \in\tau^{*}\) such that

$$F(W)\cap V \neq\phi; $$

(iv) \(F^{-} (V)\in\tau^{*}\) for every \(V \in\sigma\);

(v) \(F^{+} (K)\) is superharmonic-closed for every closed set \(K\subseteq Y\);

(vi) \(\operatorname{superharmonic-cl}(F^{+} (B)) \subseteq F^{+} (\sigma \operatorname{cl-}(B))\) for any \(B\subseteq Y\);

(vii) \(F^{-} (\operatorname{\sigma-{int}}(B))\subseteq \operatorname{superharmonic-int}(F^{-} (B))\) for any \(B\subseteq Y\);

(viii) \(\operatorname{superharmonic-fr}(F^{+} (B)) \subseteq F^{+} (\operatorname{fr}(B))\) for every \(B \subseteq Y\);

(ix) \(F:(X, \tau^{*})\rightarrow(Y, \sigma)\) is lower superharmonic-continuous.

Proof

The proof is a quite similar to that of Theorem 3.1. Recall that the net \((\chi_{i})_{(i\in l)}\) is superharmonic-convergent to \(x_{0}\), if for each \(W \in\tau^{*} (x_{O})\) there exists a \(i_{o} \in I\) such that for each \(i\ge i_{o}\) it implies \(x_{i} \in W\). □

Theorem 3.3

A superharmonic multifunction \(F : (X, \tau )\rightarrow(Y,\sigma)\) is upper superharmonic-continuous if and only if for each net \((\chi_{i})_{(i\in l)}\) superharmonic-convergent to \(x_{o}\) and for each \(V\in\sigma\) with \(F(x_{o})\subseteq V\) there is \(i_{o} \in I\) such that \(F(X_{i}) \subseteq V\) for all \(i \ge i_{o}\).

Proof

Necessity, let \(V\in\sigma\) with \(F(x_{o})\subseteq V\). By upper superharmonic-continuity of F, there is \(W\in\tau^{*}(X_{O})\) such that \(F(W)\subseteq V\). Since from the hypothesis a net \((\chi _{i})_{(i\in l)}\) is superharmonic-convergent to \(x_{o}\) and \(W \in\tau ^{*}(x_{o})\) there is one \(i_{o} \in I\) such that \(x_{i} \in W\) for all \(i > i_{o}\) and then \(F(X_{i}) \subseteq V\) for all \(i > i_{o}\). As regards sufficiency, assume the converse, i.e. there is an open set V in Y with \(F(x_{o} )\subseteq V\) such that for each \(W\in\tau^{*}\) under inclusion we have the relation \(F(W)\nsubseteq V\), i.e. there is \(x_{w} \in W \) such that \(F(x_{w}) \nsubseteq V\). Then all of \(x_{w}\) will form a net in X with directed set W of \(\tau^{*}(x_{o})\), clearly this net is superharmonic-convergent to \(x_{o}\). But \(F(x_{w})\nsubseteq V\) for all \(W \in\tau^{*}(x_{o})\). This leads to a contradiction which completes the proof. □

Theorem 3.4

A superharmonic multifunction \(F : (X,\tau )\rightarrow(Y, \sigma)\) is lower superharmonic-continuous if and only if for each \(y_{o} \in F(x_{o})\) and for every net \((\chi_{i})_{(i\in l)}\) superharmonic-convergent to \(x_{o}\), there exists a subnet \((Z_{j})_{(j\in J)}\) of the net \((\chi_{i})_{(i\in l)}\) and a net \((y_{i})_{(j,v)\in J}\) in Y so that \((y_{i})_{(j,v)\in J}\) superharmonic-convergent to y and \(y_{j} \in F(z_{j})\).

Proof

For necessity, suppose F is lower superharmonic-continuous, \((\chi_{i})_{(i\in l)}\) is a net superharmonic-convergent to \(x_{o}\), \(y \in F(x_{o})\) and \(V \in \sigma(y)\). So we have \(F(x_{o}) \cap V \ne\phi\), by lower superharmonic-continuity of F at \(x_{o}\), there is a superharmonic-open set \(W \subseteq X\) containing \(x_{o}\) such that \(W \subseteq F^{-}(V)\). We have superharmonic-convergence of a net \((\chi _{i})_{(i\in l)}\) to \(x_{0}\) and for this W, there is a \(i_{o} \in I\) such that, for each \(i > i_{o}\), we have \(x_{i} \in W\) and therefore \(x_{i} \in F^{-}(V)\). Hence, for each \(V\in\sigma(y)\), define the sets

$$I_{v} =\bigl\{ i_{o} \in I:i >i_{o} \Rightarrow x_{i} \in F^{-}(V)\bigr\} $$

and

$$J =\bigl\{ (i,V):V\in D(y),i \in I_{v}\bigr\} $$

and an order ≥ on J given as \((i^{\prime},V^{\prime}) \ge(i,V)\) if and only if \(i^{\prime}> i\) and \(V^{\prime}\subseteq V\). Also, define \(\zeta: J \rightarrow I\) by \(\zeta((j,V))= j\). Then ζ is increasing and cofinal in I, so ζ defines a subset of \((\chi_{i})_{(i\in l)}\), denoted by \((z_{i})_{(j,v)\in J}\). On the other hand for any \((j,V) \in J\), since \(j > j_{o} \) implies \(x_{j} \in F^{-}(V)\) we have \(F(Z_{j})\cap V = F(X_{j}) \cap V\ne\phi\). Pick \(y_{j} \in F(Z_{j}) \cap V \ne\phi\). Then the net \((y_{i})_{(j,v)\in J}\) is supraconvergent to y. To see this, let \(V_{0} \in\sigma(y)\); then there is \(j_{0} \in I\) with \(j_{o} = \zeta( j_{o}, V_{o} )\); \((j_{o}, V_{o}) \in J\) and \(y_{jo} \in V\). If \((j,V) > (j_{o},V_{o})\) this means that \(j > j_{o}\) and \(V \subseteq V_{o}\). Therefore

$$y_{j} \in F(z_{j}) \cap V \subseteq F(x_{j}) \cap V \subseteq F(x_{j}) \cap V_{o}. $$

So \(y_{j}\in V_{o} \). Thus \((y_{i})_{(j,v)\in J}\) is superharmonic-convergent to y, which shows the result.

To show the sufficiency, assume the converse, i.e. F is not lower superharmonic-continuous at \(x_{o}\). Then there exists \(V \in\sigma\) such that \(F(x_{o}) \cap V\ne\phi\) and for any superharmonic-neighborhood \(W \subseteq X\) of \(x_{o}\), there exists \(x_{w} \in W\) for which \(F(x_{w}) \cap V = \phi\). Let us consider the net \((\chi_{w})_{W\in \tau^{*}(\chi_{0})}\), which is obviously superharmonic-convergent to \(x_{o}\). Suppose \(y_{o} \in F(x_{o}) \cap V\), by hypothesis there is a superset \((z_{k})_{k\in K}\) of \((\chi_{w})_{W\in\tau^{*}(\chi_{0})}\) and \(y_{k} \in F(z_{k})\) like \((y_{k})_{k\in K}\) superharmonic-convergent to \(y_{o}\). As \(y_{o} \in V \in \sigma\) there is \(k_{0}^{\prime}\in K\) so that \(k>k_{0}^{\prime}\) implies \(y_{k} \in V\). On the other hand \((z_{k})_{kEK} \) is a superset of the net \((\chi^{w})_{W\in\tau ^{*}(\chi_{0})}\) and so there exists a superharmonic function \(\Omega:K \rightarrow\tau^{*}(x_{o}) \) such that \(z_{k}=\chi_{\Omega(k)}\) and for each \(W \in \tau^{*}(x_{o})\) there exists \(k_{0}^{\prime\prime}\in K\) such that \(\Omega(k_{0}^{\prime\prime}) \ge W\). If \(k\ge k_{0}^{\prime\prime}\) then \(\Omega(k) \ge\Omega(k_{0}^{\prime\prime}) \ge W \). Considering \(k_{0} \in K\) so that \(k_{o} \ge k_{0}^{\prime}\) and \(k_{o} \ge k_{0}^{\prime\prime}\). Therefore \(y_{k} \in V\) and by the meaning of the net \((\chi_{W})_{W\in\tau^{*}(\chi_{0})}\), we have

$$F(z_{k}) \cap V = F(\chi_{\Omega(K)}) \cap V = \phi. $$

This gives \(y_{k} \notin V\), which contradicts the hypothesis and so the requirement holds. □

Definition 3.2

A subset W of a space \((X, \tau)\) is called superharmonic-regular, if for any \(x \in W\) and any \(H \in\tau^{*}(x)\) there exists \(U \in\tau\) such that

$$x \in U \subseteq \operatorname{\tau-cl}(U) \subseteq H . $$

Recall that \(F: (X, \tau) \rightarrow(Y,\sigma)\) is punctually superharmonic-regular, if for each \(X\in X\), \(F(x)\) is superharmonic-regular.

Lemma 3.2

In a superharmonic space \((X,\tau)\), if \(W \subseteq X\) is superharmonic-regular and contained in a superharmonic-open set H, then there exists \(U \in\tau\) such that

$$W \subseteq U \subseteq \operatorname{\tau-cl}(U) \subseteq H . $$

For a superharmonic multifunction \(F:(X, \tau)\rightarrow(Y,\sigma)\), a superharmonic multifunction \(\operatorname{superharmonic-cl}(F):(X, \tau)\rightarrow (Y,\sigma)\) is defined as follows:

$$(\operatorname{superharmonic-cl} F) (x) =\operatorname{superharmonic-cl}\bigl(F(x)\bigr) $$

for each \(x \in X\).

Proposition 3.2

For a punctually α-paracompact and punctually superharmonic-regular superharmonic multifunction \(F: (X, \tau) \rightarrow(Y, \sigma)\), we have

$$\bigl(\operatorname{superharmonic-cl}(F)^{+} (W)\bigr) = F^{+} (W) $$

for each \(W\in \sigma^{*}\).

Proof

Let \(x \in(\operatorname{superharmonic-cl}(F))^{+}(W)\) for any \(W\in \sigma^{*}\), this means

$$F(x) \subseteq \operatorname{superharmonic-cl}\bigl(F(x)\bigr) \subseteq W, $$

which leads to \(x \in F^{+} (W)\). Hence one inclusion holds. To show the other, let \(X\in F^{+} (W)\) where \(W \in \sigma^{*} (x)\). Then \(F(x) \subseteq W\), by the hypothesis of F and the fact that \(\sigma\subseteq\sigma^{*}\), applying Lemma 3.2, there exists \(G \in\sigma\) such that

$$F(x)\subseteq G\in\operatorname{\sigma-cl}(G)\subseteq W. $$

Therefore

$$\operatorname{superharmonic-cl}\bigl(F(x)\bigr) \subseteq W. $$

This means that \(x \in(\operatorname{superharmonic-cl} F)^{+} (W)\). Hence the equality holds. □

Theorem 3.5

Let \(F (X, \tau)\rightarrow(Y, \sigma)\) be a punctually a-paracompact and punctually superharmonic-regular superharmonic multifunction. Then F is upper superharmonic-continuous if and only if

$$(\operatorname{superharmonic-cl} F): (X, \tau)\rightarrow(Y, \sigma) $$

is upper superharmonic-continuous.

Proof

As regards necessity, suppose \(V \in\sigma\) and \(x \in(\operatorname{superharmonic-cl} F)^{+} (V) = F^{+} (V)\) (see Proposition 3.2). By upper superharmonic-continuity of F, there exists \(H \in\tau^{*}(x)\) such that \(F(H) \subseteq V\). Since \(\sigma\in\sigma^{*}\), by Lemma 3.2 and the assumption of F, there exists \(G \in\sigma\) such that

$$F(h) \subseteq G \subseteq\operatorname{\sigma-cl}(G) \subseteq W $$

for each \(h \in H\).

Hence

$$\operatorname{superharmonic-cl}\bigl(F(h)\bigr) \subseteq \operatorname{superharmonic-cl} (G) \subseteq \operatorname{\sigma-cl}(G) \subseteq V $$

for each \(h \in H\), which shows that [13]

$$(\operatorname{superharmonic-cl} F) (H) \subseteq V. $$

Thus \((\operatorname{superharmonic-cl} F)\) is upper superharmonic-continuous. As regards sufficiency, assume \(V\in\sigma\) and \(X \in F^{+} (V) = (\operatorname{superharmonic-cl} F)^{+} (V)\). By the hypothesis of F in this case, there is \(H\in\tau^{*}(x)\) such that \((\operatorname{superharmonic-cl} F)(H) \subseteq V\), which obviously gives \(F(H) \subseteq V\). This completes the proof. □

Lemma 3.3

In a space \((X,\tau)\), any \(x \in X\) and \(A\subseteq X, X \in \operatorname{superharmonic-cl}(A)\) if and only if

$$A\cap W\ne\phi $$

for each \(W\in \tau^{*}(x)\).

Proposition 3.3

For a superharmonic multifunction \(F: (X, \tau) \rightarrow(Y, \sigma)\),

$$(\operatorname{superharmonic-cl} F)^{-} (W) = F^{-} (W) $$

for each \(W \in \sigma^{*}\).

Proof

Let \(x \in(\operatorname{superharmonic-cl} F)^{-} (W)\). Then

$$W \cap \operatorname{superharmonic-cl}\bigl(F(x)\bigr) \neq\phi. $$

Since \(W\in\sigma^{*}\), Lemma 3.3 gives \(W\cap F(x) \neq\phi\) and hence \(x \in F^{-}(W)\). Conversely, let \(x \in F^{-}(W)\), then

$$\phi\neq F(x)\cap W\subseteq(\operatorname{supracl} F)^{-}(x) \cap W $$

and so

$$x \in(\operatorname{superharmonic-cl} F)^{-}(W). $$

Hence

$$x \in(\operatorname{superharmonic-cl} F)^{+} (W) $$

and this completes the equality. □

Theorem 3.6

A superharmonic multifunction \(F: (X, \tau )\rightarrow(Y, \sigma)\) is lower superharmonic-continuous if and only if \((\operatorname{superharmonic-cl} F): (X, \tau) \rightarrow(Y, \sigma)\) is lower superharmonic-continuous.

Proof

This is an immediate consequence of Proposition 3.2 taking in consideration that \(\tau\subseteq\tau^{*}\) and (iv) of Theorem 3.2. □

Theorem 3.7

If \(F:(X,\tau)\rightarrow(Y, \sigma)\) is an upper superharmonic-continuous surjection and for each \(x\in X,F(x)\) is compact relative to Y. If \((X,\tau)\) is superharmonic-compact, then \((Y,\sigma)\) is compact.

Proof

Let

$$\{V_{i} : i \in I, V_{i} \in\sigma\} $$

be a cover of Y; \(F(x)\) is compact relative to Y, for each \(x \in X\). Then there exists a finite \(I_{o}(x)\) of I such that [14]

$$F(x)\subseteq U\bigl(V_{i}: i \in I_{o} (x)\bigr). $$

Upper superharmonic-continuity of F shows that there exists \(W(x) \in\tau^{*}(X,x)\) such that

$$F\bigl(W(x)\bigr) \subseteq\bigcup{V_{i}:i \in I_{o} (x)}. $$

Since \((X, \tau)\) is superharmonic-compact, there exists \({x_{1},x_{2}, \ldots,x_{n}}\) such that

$$X=\bigcup\bigl(W(x_{j}):1 \le j \le n\bigr). $$

Therefore

$$Y = F(X) = \bigcup\bigl(F\bigl(W(x_{j})\bigr): 1 \le j \le n \bigr)\subseteq\bigcup V_{i} : i \in I_{0} (X_{j}) \quad 1 \le j \le n. $$

Hence \((Y,\sigma)\) is compact. □

4 Supra-continuous superharmonic multifunctions and superharmonic-closed graphs

Definition 4.1

A superharmonic multifunction \(F:(X, \tau )\rightarrow(Y, \sigma)\) is said to have a superharmonic-closed graph if there exists \(W\in \tau^{*}(X)\) and \(H\notin\sigma^{*}(y)\) such that

$$(W\times H)\cap G(F) =\phi $$

for each pair \((x,y) \notin G(F)\).

A superharmonic multifunction \(F:(X, \tau) \rightarrow(Y, \sigma)\) is point-closed (superharmonic-closed), if for each \(x \in X\), \(F(x)\) is closed (superharmonic-closed) in Y.

Proposition 4.1

A superharmonic multifunction \(F : (X,\tau )\rightarrow(Y, \sigma)\) has a superharmonic-closed graph if and only if for each \(x \in X\) and \(y \in Y\) such that \(y \notin F(x)\), there exist two superharmonic-open sets H, W containing x and y, respectively, such that

$$F(H)\cap W = \phi. $$

Proof

As regards necessity, let \(x \in X\) and \(y \in Y\) with \(y \notin F(x)\). Then by the superharmonic-closed graph of F, there are \(H\in\tau^{*} (x)\) and \(W\in\sigma^{*}\) containing \(F(x)\) such that \((HxW)\cap G(F) = \phi\). This implies that for every \(x \in H\) and \(y \in W\) where \(y \notin F(x)\) we have \(F(H) \cap W =\phi\).

As regards sufficiency, let \((x,y) \notin G(F)\), this means \(y \notin F(x)\); then there are two disjoint superharmonic-open sets H, W containing x and y, respectively, such that \(F(H)\cap W = \phi\). This implies that \((H \times W) \cap G(F) = \phi\), which completes the proof. □

Theorem 4.1

If \(F:(X, \tau)\rightarrow(Y,\sigma)\) is an upper superharmonic-continuous and point-closed superharmonic multifunction, then \(G(F)\) is superharmonic-closed if \((Y, \sigma)\) is regular.

Proof

Suppose that

$$(x,y)\notin G(F). $$

Then \(y \notin F(x)\). Since Y is regular, there exists disjoint

$$V_{i} \in \sigma\quad (i =1,2) $$

such that

$$y \in V_{1} $$

and

$$F(x) \subseteq V_{2}. $$

Since F is upper superharmonic-continuous at x, there exists

$$W\in\tau^{*}(x) $$

such that \(F(W)\subseteq V_{2}\). As \(V_{1} \cap V_{2} = \phi\), then

$$\bigcap_{i=1}^{2}\operatorname{superharmonic-int}(V_{i})\ne\phi $$

and therefore

$$\begin{aligned}& x \in\operatorname{superharmonic-int}(W) = W, \\& y \in \operatorname{superharmonic-int}(V_{1}), \end{aligned}$$

and

$$(x, y) \in W \times \operatorname{superharmonic-int} (V_{1} ) \subseteq(X\times Y)\backslash G(F). $$

Thus

$$(X \times Y) \backslash G(F) \in \tau^{*} (X \times Y), $$

which gives the result. □

Definition 4.2

A subset W of a space \((X,\tau)\) is called α-paracompact [12] if for every open cover v of W in \((X,\tau)\) there exists a locally finite open cover ξ of W which refines v.

Theorem 4.2

Let \(F :(X, \tau)\rightarrow(Y, \sigma)\) be an upper superharmonic-continuous superharmonic multifunction from \((X,\tau)\) into a Hausdorff space \((Y,\sigma)\). If \(F(x)\) is α-paracompact for each \(x \in X\), then \(G(F)\) is superharmonic-closed.

Proof

Let \((x_{o}, y_{o}) \notin G(F)\), then \(y_{o} \notin F(x_{o})\). Since \((Y,\sigma)\) is Hausdorff, for each \(y \in F(x_{o})\) there exist \(V_{y} \in\sigma(y)\) and \(V_{y}^{*} \in \sigma(y_{o})\) such that

$$V_{y} \cap V_{y}^{*} =\phi. $$

So the family \(\{V_{y}: y\in F(x_{0})\}\) is an open cover of \(F(x_{o})\). Thus, by α-paracompactness of \(F(x_{o})\) [15], there is a locally finite open cover \(\{U_{i}:i \in I\}\) which refines \(\{V_{y}:y\in F(x_{o})\}\). Therefore, there exists \(H_{o} \in\sigma (y_{o})\) such that \(H_{o}\) intersects only finitely many members \(U_{i_{1}},U_{i_{2}},\ldots,U_{i_{n}}\) of h. Choose \(y_{1}, y_{2},\ldots,y_{n}\) in \(F(x_{o})\) such that \(U_{i_{j}}\subseteq U_{y_{j}}\) for each \(1 < j < n\), and the set

$$H= H_{o}\cap \biggl(\bigcup_{i\in I}V_{y_{i}} \biggr). $$

Then \(H \in \sigma(y_{o})\) such that

$$H \cap\biggl(\bigcup_{i\in I}V_{i}\biggr) = \phi. $$

The upper superharmonic-continuity of F means that there exists \(W \in\tau^{*}(xo)\) such that [16]

$$x_{o} \in W\subseteq F^{+}\biggl(\bigcup_{i\in I}V_{i} \biggr). $$

It follows that \((W \times H) \cap G(F)=\phi\), and hence \(G(F)\) is superharmonic-closed. □

Lemma 4.1

[14]

The following hold for \(F:(X,\tau) \rightarrow(Y,\sigma)\), \(A \subseteq X \) and \(B \subseteq Y\);

(i)

$$G_{F}^{+}(A\times B) = A \cap F^{+} (B); $$

(ii)

$$G_{F}^{-}(A\times B) = A \cap F^{-} (B). $$

Theorem 4.3

For a superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\), if GF is upper superharmonic-continuous, then F is upper superharmonic-continuous. Proof. Let \(x\in X\) and \(V\in \sigma(F(x))\). Since \(X \times V\in \tau\times\sigma\) and

$$G_{F}(x) \subseteq X \times V, $$

by Theorem  3.1, there exists \(W\in \tau^{*}(x)\) such that \(G_{F}(W)\subseteq X \times V\). Therefore, by Lemma  4.1, we get

$$W\subseteq G_{F}^{-}(X\times V) =X\cap G_{f}^{+}(V)=F^{+}(V) $$

and so \(F(W)\subseteq V\). Hence Theorem  3.1 shows also that F upper supracontinuous.

Theorem 4.4

If the graph \(G_{F}\) of a superharmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\) is lower superharmonic-continuous, then F is also.

Proof

Let \(x \in X\) and \(V \in\sigma(F(x))\) with \(F(x) \cap V \neq\phi\), also since

$$X\times V \in\tau\times\sigma, $$

then

$$G_{F} (x) \cap(X \times V) = {{x} \times F(x)}\cap(X \times V) = {x} \times\bigl(F(x) \cap V\bigr)\neq\phi. $$

Theorem 3.2 shows that there exists \(W \in\tau^{*}(x)\) such that

$$G_{F}(w) \subseteq(X \times V) \neq\phi $$

for each \(w \in W\). Hence Lemma 4.1 obtains; we have

$$W \subseteq G^{-} (X\times V) = X\cap G^{-} (V)=F^{-} (V). $$

Therefore,

$$F(w)\cap V \neq\phi $$

for each \(w \in W\) and Theorem 3.2 completes the proof. □