() @ Appl. Gen. Topol. 19, no. 1 (2018), 145-153doi:10.4995/agt.2018.7883 c© AGT, UPV, 2018 k-semistratifiable spaces and expansions of set-valued mappings Peng-Fei Yan, Xing-Yu Hu and Li-Hong Xie∗ School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P. R. China (ypengfei@sina.com, 121602580@qq.com, yunli198282@126.com) Communicated by A. Tamariz-Mascarúa Abstract In this paper, the concept of k-upper semi-continuous set-valued map- pings is introduced. Using this concept, we give characterizations of k-semistratifiable and k-MCM spaces, which answers a question posed by Xie and Yan [9]. 2010 MSC: 54C65; 54C60. Keywords: locally bounded set-valued mappings; k-MCM spaces; k- semistratifiable spaces; lower semi-continuous (l.s.c.); k-upper semi-continuous (k-u.s.c.). 1. Introduction Before stating the paper, we give some definitions and notations. For a mapping φ : X → 2Y and W ⊆ Y , the symbols φ−1[W ] and φ♯[W ] stand for {x ∈ X : φ(x) ⋂ W 6= ∅} and {x ∈ X : φ(x) ⊆ W}, respectively. A set-valued mapping φ : X → 2Y is lower semi-continuous (l.s.c) if φ−1[W ] is open in X for every open subset W of Y . Also, a set-valued mapping φ : X → 2Y is upper semi-continuous (u.s.c) if φ♯[W ] is open in X for every open subset W of Y . For mappings φ, φ ′ : X → 2Y , we express by φ ⊆ φ ′ if φ(x) ⊆ φ ′ (x) for each x ∈ X. An operator Φ assigning to each set-valued ∗Supported by NSFC(Nos. 11601393; 11526158), the PhD Start-up Fund of Natural Science Foundation of Guangdong Province (No. 2014A0303101872) Received 17 July 2017 – Accepted 05 February 2018 http://dx.doi.org/10.4995/agt.2018.7883 P.-F. Yan, X.-Y. Hu and L.-H. Xie mapping φ : X → 2Y , Φ(φ) : X → 2Y , Φ is called as a preserved order operator if Φ(φ) ⊆ Φ(φ′) whenever φ ⊆ φ′. For a space Y , define F(Y ) = {F ⊆ Y : F is a nonempty closed set in Y }. For a metric space (Y, ρ), a subset B of Y is called bounded if the diameter of B (with respect to ρ) is finite, and we define B(Y ) = {F ⊆ Y : F 6= ∅, F is closed and bounded in Y }. A sequence {Bn}n∈N of closed subsets of a space Y is called a strictly in- creasing closed cover [10] if ⋃ n∈N Bn = Y and Bn ( Bn+1 for each n ∈ N. For a space Y having a strictly increasing closed cover {Bn}, a subset B of Y is said to be bounded [10] (with respect to {Bn}) if B ⊆ Bn for some n ∈ N. Define B(Y ; {Bn}) = {F ⊆ Y : F 6= ∅, F is closed and bounded in Y }. For a space Y with a strictly increasing closed cover {Bn}, a mapping φ : X → B(Y ; {Bn}) is called locally bounded at x if there exist a bounded set V of (Y ; {Bn}) and a neighborhood O of x such that O ⊆ φ ♯[V ]; if φ is locally bounded at each x ∈ X, then φ is called locally bounded [10] on X. Let (Y, ρ) be a metric space. For a mapping φ : X → F(Y ), define Uφ = {x ∈ X : φ is locally bounded at x with respect to ρ}. Similarly, Let Y has a strictly increasing closed cover {Bn}. We also define Uφ = {x ∈ X : φ is locally bounded at x with respect to {Bn})} for a mapping φ : X → F(Y ). Clearly, Uφ is an open set in X. The insertions of functions are one of the most interesting problems in gen- eral topology and have been applied to characterize some classical cover prop- erties. For example, J. Mack characterized in [5] countably paracompact spaces with locally bounded real-valued functions as follows: Theorem 1.1 (J. Mack [5]). A space X is countably paracompact if and only if for each locally bounded function h : X → R there exists a locally bounded l.s.c. function g : X → R such that |h| ≤ g. C. Good, R. Knight and I. Stares [3] and C. Pan [6] introduced a mono- tone version of countably paracompact spaces, called monotonically countably paracompact spaces (MCP) and monotonically cp-spaces, respectively, and it was proved in [3, Proposition 14] that both these notions are equivalent. Also, C. Good, R. Knight and I. Stares [3] characterized monotonically countably paracompact spaces by the insertions of semi-continuous functions. Inspired by those results, K. Yamazaki [10] characterized MCP spaces by expansions of locally bounded set-valued mappings as follows: c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 146 k-semistratifiable spaces and expansions of set-valued mappings Theorem 1.2 (K. Yamazaki [10]). For a space X, the following statements are equivalent: (1) X is MCP; (2) for every space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each locally bounded mapping ϕ : X → B(Y ; {Bn}), a locally bounded l.s.c. mapping Φ(ϕ) : X → B(Y ; {Bn}) with ϕ ⊆ Φ(ϕ); (3) for every metric space Y , there exists a preserved order operator Φ assigning to each locally bounded set-valued mapping ϕ : X → B(Y ), a locally bounded l.s.c. set-valued mapping Φ(ϕ) : X → B(Y ) such that ϕ ⊆ Φ(ϕ); (4) there exists a preserved order operator Φ assigning to each locally bounded mapping ϕ : X → B(R), a locally bounded l.s.c. mapping Φ(ϕ) : X → B(R) such that ϕ ⊆ Φ(ϕ); (5) there exists a space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each each lo- cally bounded mapping ϕ : X → B(Y ; {Bn}), a locally bounded l.s.c. mapping Φ(ϕ) : X → B(Y ; {Bn}) such that ϕ ⊆ Φ(ϕ). Recently, Xie and Yan [9] gave the following characterizations of stratifiable and semistratifiable spaces by expansions of set-valued mappings along same lines, and asked whether there are similar characterizations for k-MCM and k-semistratifiable spaces. Theorem 1.3 (Xie and Yan [9]). For a space X, the following statements are equivalent: (1) X is stratifiable(resp. semi-stratifiable); (2) for every space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each set-valued map- ping ϕ : X → F(Y ), an l.s.c. set-valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ) is locally bounded(resp. bounded) at each x ∈ Uϕ and that ϕ ⊆ Φ(ϕ); (3) for every metric space Y , there exists a preserved order operator Φ assigning to each set-valued mapping ϕ : X → F(Y ), an l.s.c. set- valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ) is locally bounded (resp. bounded) at each x ∈ Uϕ and that ϕ ⊆ Φ(ϕ); (4) there exists a preserved order operator Φ assigning to each set-valued mapping ϕ : X → F(R), an l.s.c. set-valued mapping Φ(ϕ) : X → F(R) such that Φ(ϕ) is locally bounded (resp. bounded) at each x ∈ Uϕ and that ϕ ⊆ Φ(ϕ); (5) there exist a space Y having a strictly increasing closed cover {Bn} and a preserved order operator Φ assigning to each set-valued mapping ϕ : X → F(Y ), an l.s.c. set-valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ) is locally bounded (resp. bounded) at each x ∈ Uϕ and that ϕ ⊆ Φ(ϕ). c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 147 P.-F. Yan, X.-Y. Hu and L.-H. Xie Recently, Xie and Yan posed the following question: Question 1.4 ([9, Question 3.3]). Are there monotone set-valued expansions for k-stratifiable spaces and k-MCM along the same lines? The purposes of this paper is to attempt to answer this question by the concept of k-u.s.c set-valued mappings. Throughout this paper, all spaces are assumed to be regular, and all unde- fined topological concepts are taken in the sense given Engelking [2]. 2. Main results In this section we shall give characterization of k-MCM and k-semi stratifi- able spaces. The following concept plays an important role in this paper. Definition 2.1. For a space Y with a strictly increasing closed cover {Bn}, a mapping φ : X → B(Y ; {Bn}) is called k-upper semi-continuous (k-u.s.c.) if for every compact subset K of X, φ(K) is bounded. Obviously, for every space Y with a strictly increasing closed cover {Bn} satisfying Bn ⊂ Int Bn+1 and mapping φ : X → B(Y ; {Bn}): φ is u.s.c ⇒ φ is locally bounded ⇒ φ is k-u.s.c.. Firstly, we shall give the characterization of k-MCM by expansion of set- valued mappings. Peng and Lin gave the kβ characterization as following. They renamed the kβ as k-MCM in [7]. Proposition 2.2 ([7]). For a space X, the following statements are equivalent: (1) X is k-MCM; (2) there is an operator U assigning to a decreasing sequence of closed sets (Fj)j∈N with ⋂ j∈N Fj = ∅, a decreasing sequence of open sets (U(n, (Fj)))n∈N such that (i) Fn ⊆ U(n, (Fj)) for each n ∈ N; (ii) for any compact subset K in X, there is n0 ∈ N such that U(n0, (Fj)) ⋂ K = ∅; (iii) given two decreasing sequences of closed sets (Fj)j∈N and (Ej)j∈N such that Fn ⊆ En for each n ∈ N and that ⋂ j∈N Fj = ⋂ j∈N Ej = ∅, then U(n, (Fj)) ⊆ U(n, (Ej)), for each n ∈ N. Theorem 2.3. For a space X, the following statements are equivalent: (1) X is k-MCM; (2) for every space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each locally bounded set-valued mapping ϕ : X → F(Y ), an l.s.c. and k-u.s.c. set-valued mapping Φ(ϕ) : X → F(Y ) such that ϕ ⊆ Φ(ϕ); (3) for every metric space Y , there exists a preserved order operator Φ assigning to each locally bounded set-valued mapping ϕ : X → F(Y ), an l.s.c and k-u.s.c set-valued mapping Φ(ϕ) : X → F(Y ) such that ϕ ⊆ Φ(ϕ); c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 148 k-semistratifiable spaces and expansions of set-valued mappings (4) there exists a preserved order operator Φ assigning to each locally bounded set-valued mapping ϕ : X → F(R), an l.s.c. and k-u.s.c. set-valued mapping Φ(ϕ) : X → F(R) such that ϕ ⊆ Φ(ϕ); (5) there exists a space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each locally bounded set-valued mapping ϕ : X → F(Y ), an l.s.c. and k-u.s.c. set-valued mapping Φ(ϕ) : X → F(Y ) such that ϕ ⊆ Φ(ϕ). Proof. The implications of (2)⇒(3)⇒(4)⇒ (5) are trivial. (1)⇒ (2). Assume that X is a k-MCM space. Then there exists an operator U satisfying (i), (ii) and (iii) in Proposition 2.2. Let Y be a space having a strictly increasing closed cover {Bn}. For each locally bounded set-valued mapping ϕ : X → F(Y ) and each n ∈ N, define Fn,ϕ = {x ∈ X : ϕ(x) * Bn}. Then we have that ⋂ n∈N Fn,ϕ = ∅. Indeed, since ϕ is locally bounded, for each x ∈ X there exist an open neighborhood V of x and some i ∈ N such that ϕ(y) ⊆ Bi for each y ∈ V , which implies that V ∩ Fi,ϕ = ∅. It implies that x /∈ Fi,ϕ and ⋂ n∈N Fn,ϕ = ∅. Define Φ(ϕ) : X → F(Y ) as follows: Φ(ϕ)(x) = B1 whenever x ∈ X − U(1, (Fn,ϕ)), Φ(ϕ)(x) = Bi+1 whenever x ∈ U(i, (Fn,ϕ)) − U(i + 1, (Fn,ϕ)). Then, Φ(ϕ) is lower semi-continuous. To see this, let W be an open subset of Y and put k = min {i ∈ N : W ∩ Bi 6= ∅}. Then, one can easily check that (Φ(ϕ))−1[W ] = U(k − 1, (Fn,ϕ)) (we set U(0, (Fn,ϕ)) = X). This implies that Φ(ϕ) is lower semi-continuous. Let K be a compact subset of X, then there exists k ∈ N such that K ⋂ U(k+ 1, (Fn,ϕ)) = ∅. It implies that Φ(ϕ)(K) ⊂ Bk+1. Hence Φ(ϕ) is k-upper semi- continuous. To show that ϕ ⊆ Φ(ϕ). For each x ∈ X, there exists some i ∈ N such that x ∈ U(i − 1, (Fn,ϕ)) \ U(i, (Fn,ϕ))(we set U(0, (Fn,ϕ)) = X). Since x /∈ U(i, (Fn,ϕ)), we have x /∈ Fi,ϕ. Hence, ϕ(x) ⊆ Bi = Φ(ϕ)(x). This completes the proof of ϕ ⊆ Φ(ϕ). Finally, to show that Φ is order-preserving, let ϕ, ϕ′ : X → F(Y ) be set- valued mappings such that ϕ ⊆ ϕ′. Then, Fi,ϕ ⊆ Fi,ϕ′ for each i ∈ N, and therefore, by (iii) of Proposition 2.2, we have U(i, (Fn,ϕ)) ⊆ U(i, (Fn,ϕ′)) for each i ∈ N. For each x ∈ X. Then, Φ(ϕ′)(x) = Bk′ for some k ′ ∈ N. This implies that x ∈ U(k′ − 1, (Fn,ϕ′)) \ U(k ′, (Fn,ϕ′)). Similarly, Φ(ϕ)(x) = Bk for some k ∈ N and x ∈ U(k − 1, (Fn,ϕ)) \ U(k, (Fn,ϕ)). Clearly, k ≤ k ′. Hence, Φ(ϕ)(x) = Bk ⊆ Bk′ = Φ(ϕ ′)(x). This completes the proof of Φ(ϕ) ⊆ Φ(ϕ′) whenever ϕ ⊆ ϕ′. (5) ⇒ (1). Let Y be a space having a strictly increasing closed cover {Bn} possessing the property in (5). Let (Fj)j∈N be a sequence of decreasing closed subsets of X with ⋂ j∈N Fj = ∅. Define a set-valued mapping ϕ(Fj) : X → F(Y ) as follows: ϕ(Fj)(x) = B0 whenever x ∈ X − F1, ϕ(Fj)(x) = Bi+1 whenever x ∈ Fi − Fi+1. Then, ϕ(Fj) is locally bounded. By the assumptions, there exists a preserved operator Φ assigning to each ϕ(Fj ), an l.s.c. and k- u.s.c set-valued mapping Φ(ϕ(Fj)) : X → F(Y ) such that ϕ(Fj) ⊆ Φ(ϕ(Fj )). c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 149 P.-F. Yan, X.-Y. Hu and L.-H. Xie For every n ∈ N, define U(n, (Fj)) = X − (Φ(ϕ(Fj ))) ♯[Bn] It suffices to show the operator U satisfies (i), (ii) and (iii) of Proposition 2.2 Since ϕ(Fj) ⊆ Φ(ϕ(Fj )), for each n ∈ N we have Fn ⊆ X \ (ϕ(Fj)) ♯[Bn] ⊆ X \ (Φ(ϕ(Fj ))) ♯[Bn] = U(n, (Fj)). In addition, Φ(ϕ(Fj)) is lower semi-continuous, so U(n, (Fj)) is an open set of X for each n ∈ N. This shows that the condition (i) is satisfied. For each x ∈ X, Φ(ϕ(Fj))(x) is bounded, so there exists some n0 ∈ N such that x ∈ (Φ(ϕ(Fj ))) ♯[Bn0]. It implies that x /∈ U(n0, (Fj)). Hence,⋂ n∈N U(n, (Fj)) = ∅. Let K be a compact subset of X, then Φ(ϕ(Fj ))(K) is bounded. There exists some k0 ∈ N such that K ⊂ (Φ(ϕ(Fj))) ♯[Bk0]. It implies that K ⋂ U(k0, (Fj)) = ∅. Finally, we show the operator satisfies (iii). Let (Fj)j∈N and (F ′ j)j∈N be sequences of decreasing closed subsets of X such that Fj ⊆ F ′ j for each j ∈ N. Then one can easily show that ϕ(Fj) ⊆ ϕ(F ′j ), hence by the assumption, we have Φ(ϕ(Fj )) ⊆ Φ(ϕ(F ′j )). Therefore, U(n, (Fj)) = X \ (Φ(ϕ(Fj))) ♯[Bn] ⊆ X \ (Φ(ϕ(F ′ j ))) ♯[Bn] = U(n, (F ′ j)) holds for each n ∈ N. Thus, X is a k-MCM space. � Next, we consider the k-semi-stratifiable space. Definition 2.4. A space X is said to be semi-stratifiable [1], if there is an operator U assigning to each closed set F , a sequence of open sets U(F) = (U(n, F))n∈N such that (1) F ⊆ U(n, F) for each n ∈ N; (2) if D ⊆ F , then U(n, D) ⊆ U(n, F) for each n ∈ N; (3) ⋂ n∈N U(n, F) = F . X is said to be k-semi-stratifiable [4], if, in addition, (3′) obtained from (3) by requiring (3) a further condition ‘if a compact set K such that K ⋂ F = ∅, there is some n0 ∈ N such that K ⋂ U(n0, F) = ∅’. The following result was proved in [8]. For the completeness, we give its proof. Proposition 2.5. For any topological space X, the following statements are equivalent: (1) space X is k-semistratifiable; (2) there is an operator U assigning to a decreasing sequence of closed sets (Fj)j∈N, a decreasing sequence of open sets (U(n, (Fj)))n∈N such that (i) Fn ⊆ U(n, (Fj)) for each n ∈ N; c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 150 k-semistratifiable spaces and expansions of set-valued mappings (ii) for any compact subset K in X, if ⋂ n∈N Fn ∩ K = ∅, there is n0 ∈ N such that U(n0, (Fj)) ∩ K = ∅; (iii) Given two decreasing sequences of closed sets (Fj)j∈N and (Ej)j∈N such that Fn ⊆ En for each n ∈ N, then U(n, (Fj)) ⊆ U(n, (Ej)) for each n ∈ N. Proof. (1) ⇒ (2) Let U0 be an operator having the properties: (1), (2) and (3 ′ ) in Definition 2.4. Given any decreasing sequences of closed sets (Fj)j∈N, we can define an operator U by U((Fj)) = (U(n, (Fj)))n∈N, where U(n, (Fj)) = U0(n, Fn) for each n ∈ N. We shall prove that the operator U has the properties (i)-(iii) in (2). Because of U0 having properties (i) and (ii) in Definition 2.4, one can easily verify that U has the properties (i) and (iii) in (2). We show that the property (ii) in (2) holds for U. Take any decreasing sequences of closed sets (Fn)n∈N and any compact subset K in X such that ⋂ n∈N Fn ∩K = ∅. Then, there exists n0 ∈ N such that Fn0 ∩ K = ∅. Since X is k-semi-stratifiable, there is i ∈ N such that U0(i, Fn0)∩K = ∅. If i < n0, we have U(n0, (Fn))∩K = U0(n0, Fn0)∩K = ∅; If i ≥ n0, we also have U(i, (Fn))∩K = U0(i, Fi)∩K = ∅. Hence the operator U holds for (ii). (2) ⇒ (1) Let U0 be an operator having the properties (i)-(iii) in (2). Given any closed set F in X by letting Fn = F for each n ∈ N, we can define an operator U by U(j, F) = U0(j, (Fn)) where (U0(j, (Fn)))j∈ω = U0((Fn)). One can easily verify that the operator U has the properties in Definition 2.4. � Theorem 2.6. For a space X, the following statements are equivalent: (1) X is k-semistratifiable; (2) for every space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each set-valued map- ping ϕ : X → F(Y ), an l.s.c. set-valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ)|Uϕ is k-u.s.c. and ϕ ⊆ Φ(ϕ) ; (3) for every metric space Y , there exists a preserved order operator Φ assigning to each set-valued set-valued mapping ϕ : X → F(Y ), an l.s.c set-valued set-valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ)|Uϕ is k-u.s.c. and ϕ ⊆ Φ(ϕ); (4) there exists an order-preserving operator Φ assigning to each set-valued set-valued mapping ϕ : X → F(R), an l.s.c. set-valued mapping Φ(ϕ) : X → F(R) such that Φ(ϕ)|Uϕ is k-u.s.c and ϕ ⊆ Φ(ϕ); (5) there exists a space Y having a strictly increasing closed cover {Bn}, there exists a preserved order operator Φ assigning to each set-valued set-valued mapping ϕ : X → F(Y ), an l.s.c set-valued mapping Φ(ϕ) : X → F(Y ) such that Φ(ϕ)|Uϕ is k-u.s.c. and ϕ ⊆ Φ(ϕ). c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 151 P.-F. Yan, X.-Y. Hu and L.-H. Xie Proof. The implications of (2)⇒(3)⇒(4)⇒ (5) are trivial. (1) ⇒ (2). Assume that X is a k-semistratifiable space. Then there exists an operator U satisfying (i), (ii) and (iii) in Proposition 2.5. Let Y be a space having a strictly increasing closed cover {Bn}. For each set-valued mapping ϕ : X → F(Y ) and each n ∈ N, define Fn,ϕ = {x ∈ X : ϕ(x) /∈ Bn}. Then we have Uϕ = X\ ⋂ n∈N Fn,ϕ. Indeed, for each x ∈ Uϕ, then there exists an open neighborhood V of x and some i ∈ N such that ϕ(y) ⊆ Bi for each y ∈ V , which implies that V ⋂ Fi,ϕ = ∅. It implies that Uϕ ⊆ X − ⋂ n∈N Fn,ϕ. On the other hand, take any y ∈ X − ⋂ n∈N Fn,ϕ. Then there is Fj,ϕ such that y /∈ Fj,ϕ, and therefore, there exists an open neighborhood V of y such that V ∩ {x ∈ X : ϕ(x) * Bj} = ∅. It implies that y ∈ V ⊆ Uϕ. Define Φ(ϕ) : X → F(Y ) as follows: Φ(ϕ)(x) = B0 whenever x ∈ X − U(0, (Fn,ϕ)), Φ(ϕ)(x) = Bi+1 whenever x ∈ U(i, (Fn,ϕ))−U(i+1, (Fn,ϕ)), Φ(ϕ)(x) = Y if x ∈ X − Uϕ. Then, Φ(ϕ) is lower semi-continuous and ϕ ⊆ Φ(ϕ). We only need to show that Φ(ϕ)|Uϕ is k-u.s.c. Let K be a compact subset of Uϕ. By Proposition 2.5, there exists k ∈ N such that K ⋂ U(k + 1, (Fn,ϕ)) = ∅. It implies that Φ(ϕ)(K) ⊆ Bk+1. (5) ⇒ (1). Let Y be a space having a strictly increasing closed cover {Bn} possessing the property in (5). Let (Fj)j∈N be a sequence of decreasing closed subsets of X. Define a set-valued mapping ϕ(Fj ) : X → F(Y ) as follows: ϕ(Fj)(x) = B1 whenever x ∈ X − F1, ϕ(Fj)(x) = Bi+1 whenever x ∈ Fi − Fi+1, ϕ(Fj)(x) = Y if x ∈ X − ⋂ i∈N Fi. By the assumptions, there exists a preserved operator Φ assigning to each ϕ(Fj), an l.s.c set-valued mapping Φ(ϕ(Fj)) : X → F(Y ) such that Φ(ϕ)|Uϕ(Fj ) is k-u.s.c. and ϕ(Fj ) ⊆ Φ(ϕ(Fj )). For every n ∈ N, define U(n, (Fj)) = X − (Φ(ϕ(Fj ))) ♯[Bn]. It suffices to show the operator U satisfies (i), (ii) and (iii) of Proposition 2.5. The proof that the operator U satisfies (i) and (iii) of Proposition 2.5 is as same as Theorem 2.3, so we only shows that the operator U satisfies (ii) of Proposition 2.5. Let K be a compact subset of X satisfying K∩( ⋂ n∈N Fn) = ∅, then K ⊆ Uϕ. There exists k ∈ N such that Φ(ϕ(Fj))(K) ⊆ Bk. Hence K ∩ U(k, (Fj)) = ∅. Thus, X is a k-semistratifiable space. � Acknowledgements. We wish to thank the referee for the detailed list of corrections, suggestions to the paper, and all her/his efforts in order to improve the paper. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 152 k-semistratifiable spaces and expansions of set-valued mappings References [1] G. D. Creede, Semi-stratifiable, in: Proc Arizona State Univ Topological Conf. (1967, 1969), 318–323. [2] R. Engelking, General topology, Revised and completed edition, Heldermann Verlag, 1989. [3] C. Good, R. Knight and I. 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