@ Appl. Gen. Topol. 20, no. 1 (2019), 75-79doi:10.4995/agt.2019.9817 c© AGT, UPV, 2019 On monotonous separately continuous functions Yaroslav I. Grushka Department of Nonlinear analysis, Institute of Mathematics NAS of Ukraine, Kyiv (grushka@imath.kiev.ua) Communicated by O. Valero Abstract Let T = (T, ≤) and T1 = (T1, ≤1) be linearly ordered sets and X be a topological space. The main result of the paper is the following: If function f(t, x) : T × X → T1 is continuous in each variable (“t” and “x”) separately and function f x (t) = f(t, x) is monotonous on T for every x ∈ X, then f is continuous mapping from T × X to T1, where T and T1 are considered as topological spaces under the order topology and T × X is considered as topological space under the Tychonoff topology on the Cartesian product of topological spaces T and X. 2010 MSC: 54C05. Keywords: separately continuous mappings; linearly ordered topological spaces; Young’s theorem. 1. Introduction In 1910 W.H. Young had proved the following theorem. Theorem A (see [9]). Let f : R2 → R be separately continuous. If f(·, y) is also monotonous for every y, then f is continuous. In 1969 this theorem was generalized for the case of separately continuous function f : Rd → R (d ≥ 2): Theorem B (see [5]). Let f : Rd+1 → R (d ∈ N) be continuous in each variable separately. Suppose f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d). Then f is continuous on Rd+1. Note that theorems A and B were also mentioned in the overview [2]. In the papers [6,7] authors investigated functions of kind f : T × X → R, where Received 12 March 2018 – Accepted 10 September 2018 http://dx.doi.org/10.4995/agt.2019.9817 Ya. I. Grushka (T, ≤) is linearly ordered set equipped by the order topology, (X, τX) is any topological space and the function f is monotonous relatively to the first vari- able as well continuous (or quasi-continuous) relatively to the second variable. In particular in [7] it was proven that each separately quasi-continuous and monotonous relatively to the first variable function f : R × X → R is quasi- continuous relatively to the set of variables. The last result may be considered as the abstract analog of Young’s theorem (Theorem A) for separately quasi- continuous functions. However, we do not know any direct generalization of Theorem A (for sep- arately continuous and monotonous relatively to the first variable function) in abstract topological spaces at the present time. In the present paper we prove the generalization of theorems A and B for the case of (separately continu- ous and monotonous relatively to the first variable) function f : T × X → T1, where (T, ≤), (T1, ≤1) are linearly ordered sets equipped by the order topology and X is any topological space. 2. Preliminaries Let T = (T, ≤) be any linearly (ie totally) ordered set (in the sense of [1]). Then we can define the strict linear order relation on T such that for any t, τ ∈ T the correlation t < τ holds if and only if t ≤ τ and t 6= τ. Together with the linearly ordered set T we introduce the linearly ordered set T±∞ = (T ∪ {−∞, +∞} , ≤) , where the order relation is extended on the set T ∪ {−∞, +∞} by means of the following clear conventions: (a): −∞ < +∞; (b): (∀t ∈ T) (−∞ < t < +∞). Recall [1] that every such linearly ordered set T = (T, ≤) can be equipped by the natural “internal” order topology Tpi [T], generated by the base consisting of the open sets of kind: (τ1, τ2) = {t ∈ T | τ1 < t < τ2} ,(2.1) where τ1, τ2 ∈ T ∪ {−∞, +∞} , τ1 < τ2. Let (X, τX), (Y, τY ) and (Z, τZ) be topological spaces. By C(X, Y ) we denote the collection of all continuous mappings from X to Y . For a mapping f : X × Y → Z and a point (x, y) ∈ X × Y we write f x(y) := fy(x) := f(x, y). Recall [3] that the mapping f : X × Y → Z is refereed to as separately continuous if and only if f x ∈ C(Y, Z) and fy ∈ C(X, Z) for every point (x, y) ∈ X × Y (see also [6–8]). The set of all separately continuous mappings f : X × Y → Z is denoted by CC (X × Y, Z) [3,6–8]. Let T = (T, ≤) and T1 = (T1, ≤1) be linearly ordered sets. We say that a function f : T → T1 is non-decreasing (non-increasing) on T if and only if for every t, τ ∈ T the inequality t ≤ τ leads to the inequality f(t) ≤1 f(τ) c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 76 On monotonous separately continuous functions (f(τ) ≤1 f(t)) correspondingly. Function f : T → T1, which is non-decreasing or non-increasing on T is called by monotonous. 3. Main Results Let (X1, τX1), . . . , (Xd, τXd) (d ∈ N) be topological spaces. Further we consider X1 × · · · × Xd as a topological space under the Tychonoff topology τX1×···×Xd on the Cartesian product of topological spaces X1, . . . ,Xd. Recall [4, Chapter 3] that topology τX1×···×Xd is generated by the base of kind: { U1 × · · · × Ud | (∀j ∈ {1, . . . , d}) ( Uj ∈ τXj )} . Theorem 3.1. Let T = (T, ≤) and T1 = (T1, ≤1) be linearly ordered sets and (X, τX) be a topological space. If f ∈ CC (T × X, T1) and function fx(t) = f(t, x) is monotonous on T for every x ∈ X, then f is continuous mapping from the topological space (T × X, τ T×X) to the topological space (T1, Tpi [T1]). Proof. Consider any ordered pair (t0, x0) ∈ T × X. Take any open set V ⊆ T1 such that f (t0, x0) ∈ V . Since the sets of kind (2.1) form the base of topology Tpi [T1], there exist τ1, τ2 ∈ T1 ∪ {−∞, +∞} such that τ1 <1 f (t0, x0) <1 τ2 and (τ1, τ2) ⊆ V , where <1 is the strict linear order, generated by (non-strict) order ≤1 (on T1 ∪ {−∞, +∞}). The function f is separately continuous. So, since the sets of kind (2.1) form the base of topology Tpi [T] , there exist t1, t2 ∈ T ∪ {−∞, +∞} such that t1 < t0 < t2 and(3.1) f [(t1, t2) × {x0}] ⊆ (τ1, τ2) .(3.2) Further we need the some additional denotations. • In the case, where (t1, t0) 6= ∅ we choose any element α1 ∈ T such that t1 < α1 < t0 and denote α̃1 := α1. In the opposite case we denote α1 := t0, α̃1 := t1. • In the case (t0, t2) 6= ∅ we choose any element α2 ∈ T such that t0 < α2 < t2 and denote α̃2 := α2. In the opposite case we denote α2 := t0, α̃2 := t2. It is not hard to verify, that in the all cases the following conditions are per- formed: α1, α2 ∈ T, α̃1, α̃2 ∈ T ∪ {−∞, +∞} ; α1 ≤ α2; α̃1 < α̃2; [α1, α2] ⊆ (t1, t2) , where [α1, α2] = {t ∈ T | α1 ≤ t ≤ α2} ;(3.3) t0 ∈ (α̃1, α̃2) ⊆ [α1, α2] .(3.4) According to (3.3), α1, α2 ∈ (t1, t2). Hence, according to (3.2), interval (τ1, τ2) is an open neighborhood of the both points f (α1, x0) and f (α2, x0). c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 77 Ya. I. Grushka Since the function f is separately continuous on T × X, then there exist an open neighborhood U ∈ τX of the point x0 (in the space X) such that: f [{α1} × U] ⊆ (τ1, τ2) ;(3.5) f [{α2} × U] ⊆ (τ1, τ2) .(3.6) The set (α̃1, α̃2)×U is an open neighborhood of the point (t0, x0) in the topology τ T×X of the space T × X. Now our aim is to prove that (3.7) ∀ (t, x) ∈ (α̃1, α̃2) × U (f (t, x) ∈ (τ1, τ2) ⊆ V ) . So, chose any point (t, x) ∈ (α̃1, α̃2) × U. According to the condition (3.4), we have (t, x) ∈ [α1, α2] × U, that is α1 ≤ t ≤ α2 and x ∈ U. In accordance with (3.5), (3.6), we have f (α1, x) ∈ (τ1, τ2) and f (α2, x) ∈ (τ1, τ2). Hence, since the function fx(·) = f(·, x) is monotonous on T and α1 ≤ t ≤ α2, we deduce f (t, x) ∈ (τ1, τ2) ⊆ V . Thus, the correlation (3.7) is proven. Hence, the function f is continuous in (every) point (t0, x0) ∈ T × X. � Theorem A is a consequence of Theorem 3.1 in the case T = X = R, where R is considered together with the usual linear order on the field of real numbers and usual topology. Corollary 3.2. Let T0 = (T0, ≤0), T1 = (T1, ≤1), . . . , Td = (Td, ≤d) (d ∈ N) be linearly ordered sets, and (X, τX) be a topological space. If the function f : T1 × · · · × Td × X → T0 is continuous in each variable separately and f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d) then f is a continuous mapping from the topological space (T1 × · · · × Td × X, τ T1×···×Td×X) to the topological space (T0, Tpi [T0]). Proof. We will prove this corollary by induction. For d = 1 the corollary is true by Theorem 3.1. Assume, that the corollary is true for the number d − 1, where d ∈ N, d ≥ 2. Suppose, that function f : T1 × · · · × Td × X → T0 is satisfying the conditions of the corollary. Then we may consider this function as a mapping from T1 ×X(d) to T0, where X(d) = T2 ×· · ·×Td ×X. According to inductive hypothesis, function f (t1, ·) is continuous on X(d) for every fixed t1 ∈ T1. So f is a separately continuous mapping from T1 × X(d) to T0. Moreover, f is monotonous relatively to the first variable (by conditions of the corollary). Hence, by Theorem 3.1, f is continuous on T1 × X(d). � Theorem B is a consequence of Corollary 3.2 in the case T0 = T1 = · · · = Td = X = R, where R is considered together with the usual linear order on the field of real numbers and usual topology. In the case T0 = R, Tj = (aj, bj), X = (ad+1, bd+1) where aj, bj ∈ R and aj < bj (j ∈ {1, . . . , d + 1}) and inter- vals (aj, bj) are considered together with the usual linear order and topology, induced from the field of real numbers, we obtain the following corollary. Corollary 3.3. If the function f : (a1, b1) × · · · × (ad, bd) × (ad+1, bd+1) → R (d ∈ N) is continuous in each variable separately and f (t1, . . . , td, τ) is monotonous in each ti separately (1 ≤ i ≤ d) then f is a continuous mapping from (a1, b1) × · · · × (ad+1, bd+1) to R. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 78 On monotonous separately continuous functions Remark 3.4. In fact in the paper [5] the more general result was formulated, in comparison with Theorem B. Namely the author of [5] had considered the real valued function f (t1, . . . , td, τ) defined on an open set G ⊆ R d+1, d ∈ N such that f is continuous in each variable separately and monotonous in each ti separately (1 ≤ i ≤ d). But this result of [5] can be delivered from Corollary 3.3, because for each point t = (t1, . . . , td, τ) ∈ G in the open set G there exists the set of intervals (a1, b1) , . . . , (ad+1, bd+1) such that t ∈ (a1, b1)× · · · × (ad+1, bd+1) ⊆ G. References [1] G. 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