@ Appl. Gen. Topol. 21, no. 1 (2020), 71-79 doi:10.4995/agt.2020.12042 c© AGT, UPV, 2020 New topologies between the usual and Niemytzki Dina Abuzaid a, Maha Alqahtani b and Lutfi Kalantan a a Department of Mathematics, King Abdulaziz University, Saudi Arabia. (dabuzaid@kau.edu.sa, lkalantan@kau.edu.sa) b Department of Mathematics, King Khalid University, Saudi Arabia. (mjobran@kku.edu.sa) Communicated by O. Valero Abstract We use the technique of Hattori to generate new topologies on the closed upper half plane which lie between the usual metric topology and the Niemytzki topology. We study some of their fundamental properties and weaker versions of normality. 2010 MSC: 54A10; 54D15. Keywords: generated topology; H-space; Niemytzki plane; usual metric; CC-normal; L-normal; S-normal; C-paracompactness. 1. Notations and basic definitions We use the technique of Hattori [6, 13] to generate new topologies on the closed upper half plane which lie between the usual metric topology and the Niemytzki topology. We study some of their fundamental properties and weaker versions of normality. We denote an order pair by 〈x,y〉, the set of real numbers by R, the natural numbers by N, and the rationals by Q. Let X = {〈x,y〉 ∈ R2 : y ≥ 0} and P = {〈x,y〉 ∈ R2 : y > 0}, so the x-axis is L = X\P . Denote the usual metric topology on X by U and the Niemytzki topology on X by N . For every 〈x, 0〉 ∈ L and r ∈ R, r > 0, let D(〈x, 0〉,r) be the set of all points of P inside the circle of radius r tangent to x-axis at 〈x, 0〉 and let Dr(〈x, 0〉) = D(〈x, 0〉,r)∪{〈x, 0〉}. For every 〈x,y〉 ∈ X and r > 0, let Ur(〈x,y〉) be the set of all points of X inside the circle of radius r and centered at 〈x,y〉. Recall that the Niemytzki topology N on X is generated by the following neighborhood Received 03 July 2019 – Accepted 28 October 2019 http://dx.doi.org/10.4995/agt.2020.12042 D. Abuzaid, M. Alqahtani and L. Kalantan system: For every 〈x, 0〉 ∈ L, let B(〈x, 0〉) = {Dr(〈x, 0〉) : r > 0}. For every 〈x,y〉 ∈ P , let B(〈x,y〉) = {Ur(〈x,y〉) : r > 0}. Observe that P as a subspace of X with the usual metric topology coincides with P as a subspace of X with the Niemytzki topology. Definition 1.1. Let A be a non-empty proper subset of the x-axis L. For each 〈a, 0〉 ∈ A, let B(〈a, 0〉) = {Ur(〈a, 0〉) : r > 0}, where Ur(〈a, 0〉) is the set of all points of X inside the circle of radius r and centered at 〈a, 0〉. For each 〈a,b〉 ∈ P , let B(〈a,b〉) = {Ur(〈a,b〉) : r > 0}. So, the points in A∪P will have the same local base as in ( X , U ). For each 〈c, 0〉 ∈ L\A, let B(〈c, 0〉) = {Dr(〈c, 0〉) : r > 0}. So, the points in L\A will have the same local base as in ( X , N ). We call the topology on X generated by the neighborhood system {B(〈x,y〉) : 〈x,y〉 ∈ X} the H-generated topology on X from U and N and denote it by UAN . We call X with this H-generated topology an H-space and denote it by ( X , UAN ). Observe that if A = ∅, then UAN is the Niemytzki topology, if A is the x-axis L, then UAN is the usual topology. From now on, when we consider X with an H-generated topology UAN we are assuming that A is a non-empty proper subset of the x-axis L. Let us interchange the local bases in Definition 1.1 as follows: Let A be a non-empty proper subset of the x-axis L. For each 〈a, 0〉 ∈ A, let B(〈a, 0〉) = {Dr(〈a, 0〉) : r > 0}. For each 〈a,b〉 ∈ P , let B(〈a,b〉) = {Ur(〈a,b〉) : r > 0}. So, the points in A∪P will have the same local base as in ( X , N ). For each 〈c, 0〉 ∈ L\A, let B(〈c, 0〉) = {Ur(〈c, 0〉) : r > 0}, where Ur(〈c, 0〉) is the set of all points of X inside the circle of radius r and centered at 〈c, 0〉. So, the points in L \ A will have the same local base as in ( X , U ). We call the topology on X generated by the neighborhood system {B(〈x,y〉) : 〈x,y〉 ∈ X} the H-generated topology on X from N and U and denote it by NAU. It is clear that UAN = N(L\A)U for any subset A of the x-axis L. 2. Some Fundamental Properties Observe that for any non-empty proper subset A of the x-axis we have U ⊆UAN ⊆N . Thus ( X , UAN ) is T0, T1, Hausdorff, completely Hausdorff, and connected. To show complete regularity of ( X , UAN ) we use Frink’s theorem [4] which is the following characterization, see also [3, 1.5.G]: Theorem 2.1 (O. Frink). A space X is completely regular if and only if there exists a base B for X satisfying the following two conditions: (1) For every x ∈ X and every U ∈ B that contains x there exists V ∈ B such that x 6∈ V and U ∪V = X. (2) For any U,V ∈ B satisfying U ∪V = X, there exist U′,V ′ ∈ B such that X \V ⊆ U′, X \U ⊆ V ′, and U′ ∩V ′ = ∅. Theorem 2.2. Every H-space ( X , UAN ) is Tychonoff. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 72 New topologies between the usual and Niemytzki Proof. Denote the base of the neighborhood system {B(〈x,y〉) : 〈x,y〉 ∈ X} by B. Let W = {X \G : G ∈ B}. Define B = B∪W. Since B ⊂ B, then B is a base. We show that B satisfies the conditions of Theorem 2.1. Observe that for each G,G′ ∈ B with G ⊂ G′ we have G ⊂ G ⊂ G′. Let 〈x,y〉 ∈ X be arbitrary, and U ∈ B be arbitrary with 〈x,y〉 ∈ U. Pick K ∈ B such that 〈x,y〉 ∈ K ⊂ U. We have two cases: (1) U ∈ B. Then we can let V = X \ K. Therefore 〈x,y〉 /∈ V and U ∪V = X. (2) U ∈W. Put U = X \G; G ∈B. Here K has two cases: (a) K ∈B. Then we can let V = X \K. (b) K ∈W, where K = X\G ′ ; G′ ∈B with G ⊂ G′. Then we can let V = G′. Therefore in each case 〈x,y〉 /∈ V and U ∪V = X. So the first condition of Theorem 2.1 is satisfied. Let U,V ∈ B be arbitrary satisfying U ∪V = X. Then one and only one of the following cases is satisfied: (1) U ∈B and V ∈W, where V = X \G; G ∈B with G ⊂ U. Pick K ∈B with G ⊂ K ⊂ U. Let U′ = K and V ′ = X\K. Therefore X\V ⊂ U′, X \U ⊂ V ′, and U′ ∩V ′ = ∅. (2) U ∈W and V ∈B, where U = X \G; G ∈B with G ⊂ V . Pick K ∈B with G ⊂ K ⊂ V . Let V ′ = K and U′ = X\K. Therefore X\V ⊂ U′, X \U ⊂ V ′, and U′ ∩V ′ = ∅. (3) U ∈ W and V ∈ W, put U = X \ K, V = X \ G where K,G ∈ B with K ∩ G = ∅. Write K = Br1 (〈x,y〉) and G = Br2 (〈a,b〉). Since X = U ∪ V , then K ∩ G = ∅. Since K and G are both closed, then the distance between them are positive say δ > 0. Let �1 = r1 + δ 4 and �2 = r2 + δ 4 . Let V ′ = B�1 (〈x,y〉) and U′ = B�2 (〈a,b〉). Therefore X \V ⊂ U′, X \U ⊂ V ′, and U′ ∩V ′ = ∅. So, in all cases, the second condition of Theorem 2.1 is satisfied. Thus ( X , UAN ) is Tychonoff. � Any H-space ( X , UAN ) is first countable just by taking for each 〈x,y〉 ∈ A∪P the countable local base B′(〈x,y〉) = {U 1 n (〈x,y〉) : n ∈ N} and for each 〈x, 0〉 ∈ L\A the countable local base B′(〈x, 0〉) = {D 1 n (〈x, 0〉) : n ∈ N}. Any H-space ( X , UAN ) is separable as (Q × Q) ∩P is a countable dense subset. For the second countability, we have the following theorem: Theorem 2.3. Let ( X , UAN ) be an H-space. The following are equivalent: (1) L\A is countable. (2) ( X , UAN ) is second countable. (3) ( X , UAN ) is metrizable. Proof. (1)⇒(2) Assume that L\A is countable. Since A∪P as a subspace is metrizable, let W be a countable base for A∪P . Let B = {W,B′(〈x, 0〉) : 〈x, 0〉 ∈ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 73 D. Abuzaid, M. Alqahtani and L. Kalantan L \ A}, then B is a countable base for ( X , UAN ) because L \ A is countable. (2)⇒(3) Assume that ( X , UAN ) is second countable. Since ( X , UAN ) is also T3, see Theorem 2.2, and any T3 second countable space is metrizable [3], result follows. (3)⇒(2) Assume that ( X , UAN ) is metrizable. Since it is separable and any metrizable separable space is second countable [3, 4.1.16], result follows. (2)⇒(1) Assume that ( X , UAN ) is second countable. Suppose that L \ A is uncountable. Since any basic open set Dr(〈x, 0〉) of each element 〈x, 0〉 in L\A does not contain any element from the x-axis other than 〈x, 0〉 itself and any basic open set Ur(〈x′, 0〉) of each element 〈x′, 0〉 in A cannot be contained in Dr(〈x, 0〉), we conclude that ( X , UAN ) cannot be second countable which is a contradiction. � ( X , UAN ) need not be normal, for example, if A = {〈x, 0〉 : x < −1 or 0 < x}, then X\(A∪P) is closed uncountable discrete subspace of ( X , UAN ). Since ( X , UAN ) is also separable, then by Jones’ Lemma [7], ( X , UAN ) cannot be normal. Since any interval C in R must contain a closed bounded interval [a,b], we conclude the following theorem: Theorem 2.4. If L \ A contains a set of the form C ×{0}, where C is an interval in R, then ( X , UAN ) cannot be normal. Since any metrizable space is normal, Theorem 2.3 gives the following: Theorem 2.5. If L\A is countable, then ( X , UAN ) is normal. For each n ∈ N, Let Wn = R × [0,n). Then X ⊆ ⋃ n∈N Wn and for all n ∈ N, Wn ∈ UAN . So the family W = {Wn : n ∈ N} is a countable open cover of X which has no finite subcover. Thus ( X , UAN ) is neither compact nor countably compact. Theorem 2.6. For a non-empty proper subset A of L, ( X , UAN ) is not locally compact. Proof. Let 〈x, 0〉 ∈ L\A be arbitrary and let G be any open neighborhood of 〈x, 0〉. There exists an i ∈ R such that Di(〈x, 0〉) ⊂ Di(〈x, 0〉) = Di(〈x, 0〉) U ⊆ G. Observe that Di(〈x, 0〉) U is closed in ( X , UAN ). Now, the circumference of Di(〈x, 0〉) U contains a sequence of points which converges to 〈x, 0〉 in the usual topology, but this sequence can have no accumulation point in ( X , UAN ) as the open neighborhood Dj(〈x, 0〉), where j > i, satisfies that Dj(〈x, 0〉) does not contain any member of the sequence. Thus Dj(〈x, 0〉) U is not countably compact, hence not compact. � The first result about the Lidelöfness of an H-space is a corollary of Theorem 2.3. Theorem 2.7. If L\A is countable, then the H-space ( X ,UAN ) is Lindelöf. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 74 New topologies between the usual and Niemytzki Theorem 2.8. If A′ = {x : 〈x, 0〉 ∈ A} is dense in (R,U), then ( X ,UAN ) is Lindelöf. Proof. Let W = {Wα : α ∈ Λ} be any open cover for X. For each 〈x, 0〉 ∈ A, there exists an αx ∈ Λ such that 〈x, 0〉 ∈ Wαx. Thus for each 〈x, 0〉 ∈ A there exists rx > 0 such that 〈x, 0〉 ∈ Urx(〈x, 0〉) ⊆ Wαx. Since A′ is dense in (R,U) , then ∪x∈A′ Urx(〈x, 0〉) covers L. Now, for each 〈x,y〉 ∈ P , there exists Wα〈x,y〉 ∈ W such that 〈x,y〉 ∈ Wα〈x,y〉 . This means that there exists r〈x,y〉 > 0 such that 〈x,y〉 ∈ Ur〈x,y〉 (〈x,y〉) ⊆ Wα〈x,y〉 . Thus W ′ = {Ur〈x,y〉 (〈x,y〉),Urx(〈x, 0〉) : 〈x, 0〉 ∈ A,〈x,y〉 ∈ P} is an open cover for (X,U). Since (X,U) is Lindelöf , then there exists a countable subcover from W′. Then W has a countable open refinement . Therefore, ( X ,UAN ) is Lindelöf. � Theorem 2.9. ( X ,UAN ) is not Lindelöf if and only if L\A contains a set of the form C ×{0}, where C is an interval in R. Proof. (⇒) Assume that ( X ,UAN ) is not Lindelöf. Suppose that L\A does not contain any set of the form C ×{0}, where C is an interval in R. So, for each a,b ∈ R; a < b, there exists x ∈ A such that a < x < b. This gives that A′ = {x : 〈x, 0〉 ∈ A} is dense in (R,U). Then by Theorem 2.8 ( X ,UAN ) is Lindelöf which is a contradiction. (⇐) Assume that L\A contains a set of the form C ×{0}, where C is an interval in R. Let J = [a,b ] ⊆ C. Let � = b−a 4 . Then X ⊆ (∪〈x,y〉∈(A∪P)\([a,b ]×[0, 3� 2 )) U�2 (〈x,y〉) ) ∪ (∪〈x,0〉∈L\A D�(〈x, 0〉) ). So, {U� 2 (〈x,y〉),D�(〈x, 0〉) : 〈x,y〉 ∈ (A∪P)\([a,b ]× [0, 3�2 );〈x, 0〉 ∈ L\A} is an open cover of X which has no countable subcover because J is uncountable and for each 〈x, 0〉 ∈ J ×{0} the set D� (〈x, 0〉) contains no other element of J except 〈x, 0〉 . � 3. Other Properties of H-spaces Recall that a topological space X is called C-normal [1] (CC-normal [9], L-normal [11], S-normal [10]) if there exists a normal space Y and a bijective function f : X −→ Y such that the restriction f �A: A −→ f(A) is a homeo- morphism for each compact (countably compact, Lindelöf, separable) subspace A ⊆ X. A topological space X is called C2-paracompact [12] if there exists a T2 paracompact space Y and a bijective function f : X −→ Y such that the restriction f �A: A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. A topological space ( X , τ ) is called submetrizable if there exists a metric d on X such that the topology τd on X generated by d is coarser than τ , i.e., τd ⊆ τ , see [5]. Since submetrizabilty implies both C-normality [1] and C2-paracompactness [12], we conclude that any H-space ( X , UAN ) is both C-normal and C2-paracompact being submetrizable by the usual metric. By the theorem “If X is T3 separable L-normal and of countable tightness, then X is normal.” [11, 1.6], we obtain the following theorem: c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 75 D. Abuzaid, M. Alqahtani and L. Kalantan Theorem 3.1. ( X , UAN ) is normal if and only if it is L-normal. As ( X , UAN ) is always separable we conclude the following theorem: Theorem 3.2. ( X , UAN ) is normal if and only if it is S-normal. Now, to study the CC-normality of an H-space, we start with the CC- normality of the Niemytzki plane. We do this by three steps. Lemma 3.3. A subset C of the Niemytzki plane X is countably compact if and only if C ∩ L is finite and C is closed and bounded in X considered with its usual metric topology. Proof. Assume that C is a countably compact subspace of the Niemytzki plane X. Suppose that C ∩ L is infinite. Pick a countably infinite subset D = {〈dn, 0〉 : n ∈ N}⊆ C∩L. For each n ∈ N, consider the basic open neigh- borhood D1(〈dn, 0〉) of 〈dn, 0〉. For each 〈x,y〉 ∈ C ∩ P , consider Uy 2 (〈x,y〉) and let U = ( ⋃ 〈x,y〉∈C∩P Uy2 (〈x,y〉)) ⋃ ( ⋃ 〈x,0〉∈(C∩L)\D D1(〈x, 0〉)). Then the countable open cover {U,D1(〈dn, 0〉) : n ∈ N} of C has no finite subsover which is a contradiction. Now, Assume that C is countably compact and C∩L is finite. Suppose that C is either not closed in X considered with its usual metric topology or not bounded. Since in a metrizable space, a subspace is countably compact if and only if it compact [3, 4.1.17]. Also, in the usual metric space, a subspace is compact if and only if it is closed and bounded [3, 3.2.8]. We conclude that C is not countably compact in X considered with its usual metric topology. Since the usual metric topology U is coarser than the Niemytzki topology N we conclude that C is not countably compact in the Niemytzki plane X which is a contradiction. Conversely, assume that C ∩ L is finite and C is closed and bounded in X considered with its usual metric topology. Let W be any countable open cover for C. Then W is a countable open cover for C ∩ P. Since C is closed and bounded in X considered with its usual metric topology, then C ∩ P is closed and bounded in P considered with its usual metric topology. So, C ∩P is compact in P . Pick a finite W1, ...,Wn ∈ W such that C ∩ P ⊆ ⋃n i=1 Wi. Since C ∩ L is finite, pick for each 〈x, 0〉 ∈ C ∩ L a member Wx ∈ W such that 〈x, 0〉 ∈ Wx. Then {W1, ...,Wn,Wx : x ∈ C ∩ L} is a finite subcover. Therefore, C is countably compact. � Lemma 3.4. Let C be a subspace of the Niemytzki plane X. C is countably compact if and only if C is compact. Proof. Let C be any countably compact subspace in the Niemytzki plane X. By Lemma 3.3, C ∩ L is finite and C is closed and bounded in X considered with its usual metric topology, hence C ∩ P is compact in X with its usual metric topology [3, 3.2.8]. Let G be any open cover for C consisting of basic open sets. Since C ∩ L is finite, pick Gx ∈ G such that 〈x, 0〉 ∈ Gx. Since C ∩ P is compact, pick a finite subcover G′ of G which covers C ∩ P . Then G′ ∪{Gx : 〈x, 0〉 ∈ C ∩L} is a finite subcover of G. Thus C is compact. The other direction is clear. � c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 76 New topologies between the usual and Niemytzki Theorem 3.5. The Niemytzki plane is CC-normal. Proof. Let Y = X with its usual metric topology. Consider the identity func- tion id : X −→ Y . Since the usual metric topology U is coarser than the Niemytzki topology N , then id : X −→ Y is continuous, hence any restriction function of it is continuous. Let C be any countably compact subspace of X. By Lemma 3.4, C is compact in the Niemytzki plane, hence id|C : C −→ id(C) = C is a homeomorphism, see [3, 3.1.13]. � We use similar ideas to show that any H-space is CC-normal. Lemma 3.6. A subset C of an H-space ( X , UAN ) is countably compact if and only if C satisfies the following two conditions: (1) C is closed and bounded in ( X , U ). (2) Any infinite subset of C ∩ (L\A) has an accumulation point in C ∩A in L with its usual metric topology. Proof. Assume that C is countably compact in an H-space ( X , UAN ). Sup- pose that C is either not closed in X considered with its usual metric topology or not bounded. Since in a metrizable space, a subspace is countably compact if and only if it compact [3, 4.1.17]. Also, in the usual metric space, a subspace is compact if and only if it is closed and bounded [3, 3.2.8]. We conclude that C is not countably compact in ( X , U ). Since the usual metric topology U is coarser than the H-topology UAN we conclude that C is not countably com- pact in the H-space ( X , UAN ) which is a contradiction. Now, assume that C is a countably compact subspace of an H-space ( X , UAN ) and C is closed and bounded in ( X , U ). Suppose that there exists a countably infinite subset D = {〈dn, 0〉 : n ∈ N} of C∩(L\A) which has no accumulation point in C∩A with respect to L with its usual metric topology. For each 〈x, 0〉 ∈ C ∩A, fix rx > 0 such that Urx(〈x, 0〉) satisfies Urx(〈x, 0〉) ∩D = ∅. Let U = ( ⋃ 〈x,y〉∈C∩P Uy 2 (〈x,y〉)) ⋃ ( ⋃ 〈x,0〉∈C∩((L\A)\D) D1(〈x, 0〉)) ⋃ ( ⋃ 〈x,0〉∈C∩A Urx(〈x, 0〉)). Then the countable open cover {U,D1(〈dn, 0〉) : n ∈ N} of C has no finite subsover which is a contradiction. Conversely, let C be any subset of X satisfies the two conditions. Let W be any countable open cover for C. By condition (1), C ∩ (P ∪A) is closed and bounded in the metrizable space P ∪A. So, C ∩ (P ∪A) is compact in P ∪A. Pick a finite subcover W ′ of W for C∩(P∪A). If C∩(L\A) is finite, there exist W1, ...,Wn ∈ W such that C ∩ (L\A) ⊆ ⋃n i=1 Wi. Then W ′ ∪{W1, ...,Wn} is a finite subcover of W covers C. Now, if C ∩ (L \ A) is infinite. For each 〈x, 0〉 ∈ C ∩ (L \ A) there exists Wx ∈ W with 〈x, 0〉 ∈ Wx. Since W is countable, pick a countable subset E = {〈xn, 0〉 : n ∈ N} ⊆ C ∩ (L\A) such that C ∩ (L\A) ⊆ ⋃ n∈N Wxn. By condition (2), pick 〈y, 0〉 ∈ C ∩A such that 〈y, 0〉 is an accumulation point of E in L with its usual metric. Then the open set Wy ∈W ′ with 〈y, 0〉 ∈ Wy covers all elements of E except possibly finitely many elements say 〈xn1, 0〉, ...,〈xnm, 0〉. For each i ∈ {1, ...,m} pick Wi ∈ W c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 77 D. Abuzaid, M. Alqahtani and L. Kalantan such that 〈xni, 0〉 ∈ Wi. Then W ′ ⋃ {Wni : i ∈{1, ...,m}} is a finite subcover of W covers C. Therefore, C is countably compact. � Lemma 3.7. Let C be a subspace of an H-space ( X , UAN ). C is countably compact if and only if C is compact. Proof. Let C be any countably compact subspace in an H-space ( X , UAN ). By Lemma 3.6, C is closed and bounded in X considered with its usual metric topology, hence C ∩ (P ∪ A) is compact in X with its usual metric topol- ogy [3, 3.2.8]. Let G be any open cover for C consisting of basic open sets. Since C ∩ (P ∪ A) is compact, pick a finite subcover G′ of G which covers C ∩ (P ∪ A). In particular, G′ covers C ∩ A. Pick 〈x1, 0〉, ...,〈xn, 0〉 ∈ C ∩ A such that C ∩ A ⊆ ⋃n i=1 Uri(〈xi, 0〉). By Lemma 3.6, ⋃n i=1 Uri(〈xi, 0〉) cov- ers all possible accumulation points of C ∩ (L \ A) in L with its usual met- ric topology. Thus ⋃n i=1 Uri(〈xi, 0〉) covers all points of C ∩ (L \ A) except possibly finitely many points, say 〈y1, 0〉, ...,〈ym, 0〉 ∈ C ∩ (L \ A), because if (C ∩ (L \ A)) \ ( ⋃n i=1 Uri(〈xi, 0〉)) is infinite, then any countably infinite subset of it will not have an accumulation point in C ∩ A in L with its usual metric and this contradicts the countable compactness of C, see Lemma 3.6. For each j ∈ {1, ...,m} pick Gj ∈ G such that 〈yj, 0〉 ∈ Gj. Then G′ ∪ {Gj : j ∈ {1, ...,m}} is a finite subcover for G covers C. Thus C is compact. � Theorem 3.8. Any H-space ( X , UAN ) is CC-normal. Proof. Let Y = X with its usual metric topology. Consider the identity func- tion id : X −→ Y . Since the usual metric topology U is coarser than the H-topology UAN , then id : X −→ Y is continuous, hence any restriction function of it is continuous. Let C be any countably compact subspace in ( X , UAN ). 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Zaitsev, On certain classes of topological spaces and their bicompactifications, Dokl. Akad. Naur SSSR 178 (1968), 778–779. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 79