Maximal classes for families of lower and upper semicontinuous functions with a closed graph

In this paper we characterize the following maximal classes for families of lower and upper semicontinuous functions with a closed graph: the maximal additive class, the maximal multiplicative class and the maximal classes with respect to maximum and minimum.


Introduction
The letters R, Q and N denote the real line, the set of rationals and the set of positive integers, respectively. The family of all functions from a set X into Y is denoted by Y X . For each set A ⊂ X its characteristic function is denoted by χ A . In particular, χ ∅ stands for the zero constant function.
Let X be a topological space. The symbol X d denotes the set of all accumulation points of X . For each set A ⊂ X the symbols int A and cl A denote the interior and the closure of A, respectively. The spaces R and X × R are considered with their standard topologies. We say that a function f : X → R has a closed graph, if the graph of f , i.e., the set {(x, f (x)) : x ∈ X } is a closed subset of the product X × R. We say that a function f : X → R is lower (upper) semicontinuous at a point x ∈ X , if for each ε > 0 there is an open neighborhood U of x such that f (z) > f (x) − ε ( f (z) < f (x) + ε, respectively) for each z ∈ U . If f : X → R is lower (upper) semicontinuous at each point x ∈ X , then we say that the function f is lower (upper, respectively) semicontinuous. Let C onst(X ), C(X ), B Jolanta Kosman jola.kosman@wp.pl 1 Institute of Mathematics, Kazimirz Wielki University, pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland U(X ), lsc(X ), usc(X ) denote the class of all real-valued functions on X that are constant, continuous, have a closed graph, are lower and upper semicontinuous, respectively. Obviously C(X ) ⊂ U(X ) (see also e.g. [5]) and C(X ) = lsc(X )∩usc(X ). For F (X ) and G(X ) nonempty subsets of R X the symbol FG(X ) denotes the class F (X ) ∩ G(X ). Further denote by F + (X ) the family of all nonnegative functions from F (X ). Let f ∈ R X . The symbol G( f ) denotes the graph of f and the symbols C( f ) and D( f ) denote the sets of points of continuity and discontinuity of f , respectively. For each If F ⊂ R X is a family of functions, denote by The above classes M a (F), M m (F), M max (F) and M min (F) are called the maximal additive class for F , the maximal multiplicative class for F , the maximal class with respect to maximum and minimum for F , respectively.
In 1987 Menkyna [7] characterized the maximal additive and multiplicative classes for the family of functions with a closed graph. He proved that M a (U(X )) = C(X ) for a topological space X [7, is an open set} for a locally compact normal topological space X [7, Theorem 2]. Let Q(X ) denote the family of all quasi-continuous functions from a topological space X to R. Recall that f ∈ Q(X ) if and only if for each x ∈ X , ε > 0 and for each neighbourhood U of x there is a nonempty open set V ⊂ U such that | f (x) − f (y)| < ε for each y ∈ V . In 2008 Sieg [8] considered real functions defined on R and showed that M a (QU(R)) = C(R), M m (QU(R)) = { f ∈ C(R) : f = χ ∅ or f (x) = 0 for all x ∈ R} and M max (QU(R)) = M min (QU(R)) = ∅. In 2014 Szczuka (see [9,10]) characterized the following maximal classes for lower and upper semicontinuous strongŚwiatkowski functions and lower and upper semicontinuous extra strongŚwiatkowski functions: the maximal additive class, the maximal multiplicative class and the maximal classes with respect to maximum. She proved, among others, that if F denotes the family of lower semicontinuous strongŚwiatkowski real functions defined on R, then M a (F) = C onst [9, Theorems 3.1], M m (F) = C onst + [9, Theorem 3.2] and M max (F) = C onst [9,Theorem 3.3].
In this paper we deal with the families of lower and upper semicontinuous functions with a closed graph. We obtain the following results: is an open set and f (x) ≥ 0 for all x ∈ X } = M m (Uusc(X )), where X is a perfectly normal topological space such that X = X d (Theorems 2.7, 3.4), • M max (Ulsc(X )) = Ulsc(X ), where X is a topological space (Theorem 2.10), • M min (Uusc(X )) = Uusc(X ), where X is a topological space (Theorem 3.5), • M min (Ulsc(X )) = M max (Uusc(X )) = ∅, where X is a perfectly normal topological space such that X d = ∅ (Corollary 2.15, Theorems 3.6).

Lower semicontinuous functions with a closed graph
We start with a following proposition.

Proposition 2.1 Let X be a topological space. A function f : X → R has the closed graph if and only if for each x ∈ X and for each m ∈
Proof The implication (⇐) we can find in [2] (see p. 118, lines [11][12][13][14]. The implication (⇒) immediately follows from [6] or [1, Observe that, the equivalence of this proposition also immediately follows from [1, Proposition 2].
From above and the definitions of the class lsc we obtain:

Lemma 2.2 Let X be a topological space. A function f : X → R is lower semicontinuous function with a closed graph if and only if for each x ∈ X and for each m ∈
Proof First, assume that for each x ∈ X and for each m . This completes the proof.
The next lemma follows from Proposition 2.1 and Lemma 2.2.

Lemma 2.3 Let X be a topological space. Then
Now, we will characterize the class of the sums of lower semicontinuous functions with a closed graph.
Proof Let f, g ∈ Ulsc(X ). Fix x ∈ X and m ∈ N. Let k ∈ N be such that We consider four cases.

Theorem 2.5 Let X be a topological space. Then M a (Ulsc(X )) = Ulsc(X ).
Proof Since χ ∅ ∈ Ulsc(X ), we conclude that M a (Ulsc(X )) ⊂ Ulsc(X ). The inclusion Ulsc(X ) ⊂ M a (Ulsc(X )) follows from Lemma 2.4. Now, recall the following lemma [7, Lemma 2], which will be applied in this paper. Proposition 2.6 Let X be a topological space and let f ∈ C(X ). Then the function g : X → R defined by the formula has the closed graph. Proof We will prove this theorem in four parts. First, we will show that M m (Ulsc(X )) ⊂ C(X ). Let f ∈ M m (Ulsc(X )). Since χ R , −χ R ∈ Ulsc(X ), we have f ∈ lsc(X ) and − f ∈ lsc(X ). Consequently f ∈ lsc(X ) ∩ usc(X ) = C(X ). Now, we assume that the function f ∈ C(X ) and the set [ f = 0] is not open. We will show that f / ∈ M m (Ulsc(X )) (The proof of this part is similar to the second part of the proof of [7, Theorem 2]). Define the function g : X → R by the formula

Theorem 2.7 Let X be a normal topological space such that each singleton is G δ -set. Then
By Proposition 2.6 the function g has the closed graph. Moreover g is non-negative function and consequently, by Lemma 2.3, g ∈ Ulsc(X ). Now, we will show that f · g / ∈ Ulsc(X ). Define the function g : X → R by the formula Observe that, by Proposition 2.6 and Lemma 2.3, g ∈ Ulsc(X ).
In the last part suppose that f ∈ C(X ), the set [ f = 0] is open, [ f < 0] d = ∅ and g ∈ Ulsc(X ). Then, by [7,Theorem 2], ( f · g) ∈ U(X ) (see also the third part of the proof of [7,Theorem 2]). It is enough to show that ( f · g) ∈ lsc(X ). Let x 0 ∈ X . If f (x 0 ) ≤ 0, then the function f · g is continuous at x 0 and consequently f · g is a lower semicontinuous at this point. Indeed, if f (x 0 ) = 0, then by the assumption Since g ∈ lsc(X ), f is continuous and positive function on U , the function f · g is a lower semicontinuous at x 0 . The proof is complete.
It is easy to see that from above for X = R we have the following corollary.

Lemma 2.9 Let X be a topological space and let f, g ∈ Ulsc(X ). Then the real function h = max{ f, g} defined on X is a lower semicontinuous function with a closed graph.
Proof Let f, g ∈ Ulsc(X ). We will use Lemma 2.2. Fix x ∈ X and m ∈ N. Then there exists a neighbourhood V of x such that f (z) ∈ ( f (x) − 1/m, f (x) + 1/m) ∪ (m, ∞) and g(z) ∈ (g(x) − 1/m, g(x) + 1/m) ∪ (m, ∞) for each z ∈ V . We assume that f (x) ≥ g(x) (The case f (x) < g(x) is analogous). Then h(x) = f (x) and it is easy to see that h(z) ∈ (h(x) − 1/m, h(x) + 1/m) ∪ (m, ∞) for each z ∈ V . So, h ∈ Ulsc(X ). Theorem 2.10 Let X be a topological space. Then M max (Ulsc(X )) = Ulsc(X ).
Proof The inclusion Ulsc(X ) ⊂ M max (Ulsc(X )) follows from Lemma 2.9. So, we will only prove that M max (Ulsc(X )) ⊂ Ulsc(X ). Let f : X → R be a function such that f / ∈ Ulsc(X ). We choose x 0 ∈ X and m ∈ N, such that m ≥ f ( . This completes the proof.

Theorem 2.11
Let X be a topological space such that U(X ) = C(X ). Then M min (Ulsc(X )) = ∅.
Proof Let f ∈ R X . We will show that there is a function g ∈ Ulsc(X ) such that the function h = min{ f, g} / ∈ Ulsc(X ). Let g 1 : X → R be a function with a closed graph and let x 0 ∈ D(g 1 ). Put g 2 = |g 1 |. Then g 2 ∈ Ulsc and there is a net (x γ ) γ ∈ of elements of X which converges to the point x 0 and a net (g 2 (x γ )) γ ∈ diverges to ∞. We consider two cases.
It is easy to see that g ∈ Ulsc(X ). Let h = min{ f, g}. Then h( It is easy to see that Remark 1 Let X be a topological space such that U(X ) = C(X ). Then M min (Ulsc(X )) = C.

Upper semicontinuous functions with a closed graph
First, we recall some basic property of the functions with a closed graph [3, Proposition 2] Proposition 3.1 Let X be a topological space. Let α be a real number. If f ∈ U(X ), then α · f ∈ U(X ).
From above and the definitions of the classes lsc(X ) and usc(X ) we obtain: Proposition 3.2 Let X be a topological space. For each function f ∈ R X we have f ∈ Uusc(X ) if and only if (− f ) ∈ Ulsc(X ). Now, we will characterize the following maximal classes for the family of upper semicontinuous functions with a closed graph: the maximal additive class, the maximal multiplicative class and the maximal classes with respect to maximum and minimum. Theorem 3.3 Let X be a topological space. Then M a (Uusc(X )) = Uusc(X ).
The next theorem follows from Proposition 3.2. Proof Since − min{ f, g} = max{− f, −g} for each functions f, g ∈ R X , by Proposition 3.2, we conclude that f ∈ M min (Uusc(X )) if and only if − f ∈ M max (Ulsc(X )). Now, using Theorem 2.10 and again Proposition 3.2, we obtain that M min (Uusc(X )) = Uusc(X ).
It is easy to see that using Theorem 2.11, Remark 1 and the equivalence f ∈ M max (Uusc(X )) if and only if − f ∈ M min (Ulsc(X )), we conclude that: Theorem 3.6 Let X be a topological space. Then M max (Uusc(X )) = M min (Ulsc(X )).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.