/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 514-525 ISSN 1307-5543 – www.ejpam.com On Locally Hurewicz Spaces Clarice Aparecida Roika1, Soraya R.T Kudri1∗, Tomaz. K. Breuckmann1 1Department of Mathematics, Federal University of Paraná, P. O. Box 019081, Curitiba, PR, 81531- 990, Brazil. Abstract. In this paper we define locally Hurewicz spaces, weakly locally Hurewicz spaces and rela- tively locally Hurewicz spaces. We obtain some results and prove the equivalence of those definitions in Hausdorff C-spaces. 2010 Mathematics Subject Classifications: 54D20 Key Words and Phrases: Hurewicz Spaces, Locally Hurewicz Spaces. 1. Introduction In [3] Hurewicz introduced the notion of Hurewicz topological spaces. Those spaces gen- eralize compact spaces and are contained in the class of Lindelöf spaces. The principal purpose of this work is to localize the Hurewicz property, introducing here the locally Hurewicz spaces. In general topology there are three ways to define local compactness, which here are called local compactness, weak local compactness and relative local compactness. In this paper we define locally Hurewicz spaces, weakly locally Hurewicz spaces and relatively locally Hurewicz spaces, prove their equivalence in Hausdorff C-spaces and study some of their properties. 2. Preliminaries Throughout this paper we use the notation F ⊂<∞ X as abbreviation for "F is a finite subset of X". Definition 1. [4] A topological space 〈X , T 〉 is locally compact if and only if for each x ∈ X and for every neighborhood V of x, there are U ∈ T and a compact subset C of X such that x ∈ U ⊂ C ⊂ V . Definition 2. [5] A topological space 〈X , T 〉 is weakly locally compact if and only if for each x ∈ X there are U ∈ T and a compact subset C of X such that x ∈ U ⊂ C. ∗Corresponding author. Email address: soraya@onda.com.br (S.R.T.Kudri) http://www.ejpam.com 514 c© 2015 EJPAM All rights reserved. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 515 Definition 3. [2] A topological space 〈X , T 〉 is relatively locally compact if and only if for each x ∈ X there are U ∈ T such that x ∈ U and U is compact. Definition 4. [1] In any nonempty set X we can define a topology T by considering as open sets the empty set and all subsets of X containing a particular point p ∈ X . We shall call it the particular point p topology. Definition 5. [3] A topological space 〈X , T 〉 is Hurewicz if and only if for each sequence {Un}n∈N of open coverings of X , there exists a sequence {Vn}n∈N such that: (i) for each n ∈ N, Vn ⊂<∞ Un. (ii) ∀x ∈ X ,∃n0 ∈ N such that ∀n ∈ N if n≥ n0, there exists V ∈ Vn with x ∈ V . Example 1. The Real line R in its usual topology is a Hurewicz space. Let {Un}n∈N be a sequence of open coverings of R, where Un = {Un j} j∈Jn . Since for each n ∈ N, [−n, n] ⊂ R is compact, ∃In ⊂<∞ Jn such that Vn = {Un j} j∈In covers [−n, n], then the sequence {Vn}n∈N is such that: (i) ∀n ∈ N, In ⊂<∞ Jn and Vn = {Un j} j∈In , then we have Vn ⊂<∞ Un. (ii) For each x ∈ R, we have that there is n0 ∈ N such that |x | ≤ n0. For n ∈ N, if n≥ n0, then |x | ≤ n, i.e., x ∈ [−n, n]. But Vn = {Un j} j∈In covers [−n, n], so there exists j ∈ In such that x ∈ Un j ∈ Vn. Proposition 1. Let X = ⋃ i∈N Ki , where ∀i ∈ N, Ki is compact and Ki ⊂ Ki+1, then X is Hurewicz. Proof. Let {Un}n∈N be a sequence of open coverings of X , where each Un = {Un j} j∈Jn . Since ∀n ∈ N, Kn ⊂ X we have that Un covers Kn and by the compactness of Kn there is In ⊂<∞ Jn such that Kn ⊂ ⋃ j∈In Un j . For each n ∈ N let Vn = {Un j} j∈In , then the sequence {Vn j}n∈N is such that: (i) ∀n ∈ N, Vn ⊂<∞ Un. (ii) For each x ∈ X , since X = ⋃ i∈N Ki there exists n0 ∈ N such that x ∈ Kn0 ⊂ ⋃ j∈In0 Un0 j , then x ∈ Un0 j0 for some j0 ∈ In0 , then Un0 j0 ∈ Vn. From Ki ⊂ Ki+1 we wave that ∀n ∈ N, n≥ n0, x ∈ Kn0 ⊂ Kn ⊂ ⋃ j∈In Un j . Hence there is j ∈ In such that x ∈ Un j ∈ Vn. So X is Hurewicz. Proposition 2. Let X = ⋃ i∈N Ki , where for each i ∈ N, Ki is compact, then X is Hurewicz. Proof. Let X = ⋃ i∈N Ki . Consider for each i ∈ N, K ′i = i ⋃ j=1 K j . We have that K ′i is compact, K ′i ⊂ K ′i+1 and X = ⋃ i∈N K ′i then by the previous proposition X is Hurewicz. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 516 Corollary 1. If a topological space 〈X , T 〉 is compact, then 〈X , T 〉 is Hurewicz. Proof. It follows immediately from the Proposition 2. Proposition 3. If a topological space 〈X , T 〉 is Hurewicz, then 〈X , T 〉 is Lindelöf. Proof. Let U = {U j} j∈J be an open covering of X . Consider the sequence {Un}n∈N, where Un = U. Since X is Hurewicz, there exists a sequence {Vn}n∈N such that: (i) ∀n ∈ N, Vn ⊂<∞ Un = U, i.e., ∃In ⊂<∞ J such that Vn = {U j} j∈In . (ii) For each x ∈ X ,∃n0 ∈ N such that ∀n ∈ N if n≥ n0 then there exists V ∈ Vn with x ∈ V . Considering V = {U j; j ∈ In, n ∈ N}, we have that V is a countable subcovering of X , because for each n ∈ N, In is finite and for each x ∈ X , by (ii) there exists n0 ∈ N such that ∀n ∈ N if n ≥ n0, then there is V ∈ Vn such that x ∈ V , then there exists V ∈ Vn0 , such that x ∈ V , but from V ∈ Vn0 , we have that, V = U j , for some j ∈ In0 , then V ∈ V. So X is a Lindelöf space. Example 2. The Real line R in the particular point p topology is not a Lindelöf space. In fact, since {{x , p}; x ∈ R} is an open covering of R which does not have a countable subcovering. By the previous proposition, the Real line in the particular point p topology is not Hurewicz. Proposition 4. Let X and Y be topological spaces and let X be Hurewicz. If f : X → Y is a surjective continuous function, then Y is Hurewicz. Proof. Let {Un}n∈N be a sequence of open coverings of Y , where each Un = {Un j} j∈Jn . Consider for each n ∈ N,Wn = { f −1(Un j)} j∈Jn . By the continuity of f we have that, f −1(Un j) is open in X and we also have that Wn is a covering of X , because if x ∈ X we have that f (x) ∈ Y , hence ∃ j ∈ Jn such that f (x) ∈ Un j , then x ∈ f −1(Un j). Therefore, {Wn}n∈N is a sequence of open coverings of X . Since X is Hurewicz, there is a sequence {Hn}n∈N, such that: (i) ∀n ∈ N, Hn ⊂<∞Wn, i. e., there is In ⊂<∞ Jn, such that Hn = { f −1(Un j)} j∈In . (ii) For each x ∈ X ,∃n0 ∈ N such that ∀n ∈ N if n≥ n0, then there exists V ∈ Hn with x ∈ V , i.e., there is j ∈ In with x ∈ f −1(Un j). Consider ∀n ∈ N, Vn = {Un j} j∈In . Then the sequence {Vn}n∈N is such that: (i) ∀n ∈ N, Vn ⊂<∞ Un. (ii) For each y ∈ Y , since f is surjective ∃x ∈ X such that f (x) = y . By previous (ii) there exists n0 ∈ N, such that ∀n ∈ N if n ≥ n0, then there exists j ∈ In with x ∈ f −1(Un j). Since j ∈ In we have that y = f (x) ∈ f ( f −1(Un j)) ⊂ Un j ∈ Vn. So Y is Hurewicz. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 517 Definition 6. Let X be a topological space and Y a subset of X . We say that Y is Hurewicz if and only if Y is a Hurewicz subspace of X . Proposition 5. Let Y be a subspace of X . Y is Hurewicz if and only if for each sequence {Un}n∈N of coverings of Y by open sets in X , there is a sequence {Vn}n∈N such that: (i) ∀n ∈ N, Vn ⊂<∞ Un. (ii) ∀y ∈ Y , ∃n0 ∈ N, such that ∀n ∈ N if n≥ n0, then there exists V ∈ Vn with y ∈ V . Proof. (⇒) Considering Y Hurewicz, let {Un}n∈N be a sequence of coverings of Y by open sets in X , where ∀n ∈ N, Un = {Un j} j∈Jn . Consider the sequence {Wn}n∈N where each Wn = {Wn j} j∈Jn with Wn j = Un j ∩ Y . Then Wn is a covering of Y since for each y ∈ Y by the fact that Un be a covering of Y we have that there is j ∈ Jn with y ∈ Un j , then y ∈ Un j ∩ Y =Wn j andWn is formed by open sets in Y , but Y is Hurewicz then there exists a sequence {Hn}n∈N, such that: (i) ∀n ∈ N, Hn ⊂<∞Wn, i. e., ∃In ⊂<∞ Jn such that Hn = {Wn j} j∈Jn . (ii) For each y ∈ Y,∃n0 ∈ N such that ∀n ∈ N if n≥ n0, then there exists V ∈ Hn with y ∈ V . Consider ∀n ∈ N, Vn = {Un j} j∈In , then the sequence {Vn}n∈N is such that: (i) ∀n ∈ N, Vn ⊂<∞ Un. (ii) If y ∈ Y , from previous (ii) ∃n0 ∈ N, such that ∀n ∈ N if n≥ n0 then there exists V ∈ Hn with y ∈ V . Therefore there is j ∈ In with V =Wn j = Un j ∩ Y . Since j ∈ In we have that Un j ∈ Vn and y ∈ Un j . (⇐) Let {Un}n∈N be a sequence of coverings of Y by open sets in Y , where eachUn = {Un j} j∈Jn . Since Y is subspace of X , for each n ∈ N and for each j ∈ Jn there exists a open set Vn j in X such that Un j = Vn j∩Y . Consider ∀n ∈ N,Wn = {Vn j} j∈In , thenWn is a covering of Y because, if y ∈ Y , ∃ j ∈ Jn such that y ∈ Un j , then y ∈ Vn j and the sequence {Wn j} j∈Jn is a sequence of coverings of Y by open sets in X , then there exists a sequence {Hn}n∈N, such that: (i) ∀n ∈ N, Hn ⊂<∞Wn, i. e., ∃In ⊂<∞ Jn, such that Hn = {Vn j} j∈In . (ii) ∀y ∈ Y,∃n0 ∈ N such that ∀n ∈ N if n≥ n0 then there exists V ∈ Hn with y ∈ V . Consider ∀n ∈ N, Vn = {Un j} j∈In , then the sequence {Vn}n∈N is such that: (i) ∀n ∈ N, Vn ⊂<∞ Un. (ii) If y ∈ Y , from previous (ii) ∃n0 ∈ N, such that ∀n ∈ N if n ≥ n0 then there is V ∈ Hn with y ∈ V . Since V ∈ Hn we have that there exists j ∈ In with y ∈ Vn j , but y ∈ Y , then y ∈ Vn j ∩ Y = Un j ∈ Vn, because j ∈ In. So Y is a Hurewicz space. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 518 Proposition 6. If F is a closed subspace of a Hurewicz space X , then F is Hurewicz. Proof. Let {Un}n∈N be a sequence of coverings of F by open sets in X , where each Un = {Un j} j∈Jn . Since F is a closed set, F c is an open set. ConsiderWn = {Un j} j∈In ∪ F c , then Wn is an open covering of X . Let {Wn}n∈N be a sequence of open coverings of X . Since X is Hurewicz, there exists a sequence {Hn}n∈N, such that: (i) ∀n ∈ N, Hn ⊂<∞ Wn, i. e., there exists a finite set In, such that Hn = {Vn j} j∈In where Vn j = Uni for some i ∈ Jn or Vn j = F c . (ii) For each x ∈ X ,∃n0 ∈ N such that ∀n ∈ N if n≥ n0 then, there exists V ∈ Hn with x ∈ V . Consider Vn = {Un j ∈ Un; Un j ∈ Hn}, then the sequence {Vn}n∈N, is such that: (i) ∀n ∈ N, we have that Vn ⊂<∞ Un. (ii) Let y ∈ F . Then from previous (ii) ∃n0 ∈ N such that ∀n ∈ N if n ≥ n0, there exists V ∈ Hn with y ∈ V . Since V ∈ Hn, we have that V = Un j ∈ Un or V = F c , but since y /∈ F we have V = Uni ∈ Un, thus V ∈ Vn, therefore y ∈ V ∈ Vn. By Proposition 5 F is a Hurewicz space. Proposition 7. Let X be a topological space and let H and Y be subspaces of X , with H Hurewicz and Y closed, then H ∩ Y is Hurewicz. Proof. Let {Un}n∈N be a sequence of coverings of H ∩ Y by open sets in X , where each Un = {Un j} j∈Jn . Consider ∀n ∈ N, Hn = {Un j} j∈Jn ∪ Y c , then Hn is a covering of H by open sets in X , because Y is a closed set we have that Y c is an open set in X and for each x ∈ H if x ∈ Y then x ∈ H ∩ Y , then there exists j ∈ Jn such that x ∈ Un j , if x /∈ Y then x ∈ Y c . Therefore we have that the sequence {Hn}n∈N is a sequence of open coverings of H by open sets in X , since H is Hurewicz in X by Proposition 5, there is a sequence {Wn}n∈N such that: (i) ∀n ∈ N,Wn ⊂<∞ Hn, i. e., there exists a finite subset In such thatWn = {Vn j} j∈In where Vn j = Uni for some i ∈ Jn or Vn j = Y c . (ii) For each x ∈ H, ∃n0 ∈ N such that ∀n ∈ N if n≥ n0 then there exists V ∈Wn with x ∈ V . Consider Vn = {Un j ∈ Hn; Un j ∈ Un}, then the sequence {Vn}n∈N, is such that: (i) ∀n ∈ N, Vn ⊂<∞ Un, because Hn is formed by finite elements. (ii) For each x ∈ H ∩ Y , we have that x ∈ H and x ∈ Y . If x ∈ H, by previous (ii) ∃n0 ∈ N such that ∀n ∈ N if n≥ n0 then there exists V ∈Wn with x ∈ V . From V ∈ Hn, we have that V = Uni ∈ Un or V = Y c . Since x /∈ Y c , then V = Uni ∈ Un, hence V ∈ Vn, then x ∈ V ∈ Vn. By Proposition 5 H ∩ Y is a Hurewicz space. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 519 3. C-Space Definition 7. A topological space 〈X , T 〉 is a C-space if and only if, for each x ∈ X and for each sequence {An}n∈N, where An = {An j ∈ T ; 1 ≤ j ≤ kn} with x ∈ kn ⋂ j=1 An j , there exists V ∈ T such that ∀n ∈ N, x ∈ V ⊂ kn ⋂ j=1 An j . Example 3. Consider X 6= ;, with the discrete topology. X is a C-space, because for each x ∈ X and for each sequence {An}n∈N where for each n ∈ N, An = {An j} kn j=1 with An j open and x ∈ kn ⋂ j=1 An j , then {x} is an open set such that ∀n ∈ N, x ∈ {x} ⊂ kn ⋂ j=1 An j . Example 4. The real line R in its usual topology is not a C-space, because 0 ∈ R and considering the sequence {An}n∈N, where each An = {(− 1 k , 1 k );∀k ∈ N, 1 ≤ k ≤ n}, we have that for each n ∈ N, 0 ∈ n ⋂ k=1 (−1 k , 1 k ), but there is not an open set U in R such that ∀n ∈ N, 0 ∈ U ⊂ n ⋂ k=1 (−1 k , 1 k ). Lemma 1. Let X be a Hausdorff C-space, A a Hurewicz subspace of X and x0 /∈ A, then there are disjoint open sets V and U in X containing x0 and A, respectively. Proof. Consider a ∈ A, with x0 /∈ A we have that x0 6= a, since X is a Hausdorff space, there are Va and Ua disjoint open sets in X containing x0 and a, respectively. By considering ∀n ∈ N , Un = {Ua}a∈A, we have that {Un}n∈N is a sequence of coverings of A by open sets in X . Since, A is Hurewicz by Proposition 5 there exists a sequence {Wn}n∈N such that: (i) ∀n ∈ N,Wn ⊂<∞ Un,i. e., ∃In ⊂<∞ A, such thatWn = {Ua}a∈A. (ii) For each y ∈ A, ∃n0 ∈ N, such that ∀n ∈ N if n≥ n0, there exists a ∈ In with y ∈ Ua. Consider now, ∀n ∈ N, Vn = {Va}a∈In , since for each a ∈ A, x0 ∈ Va then ∀n ∈ N, x0 ∈ ⋂ a∈In Va, but since X is a C-space we have that there exists an open set V in X such that ∀n ∈ N, x0 ∈ V ⊂ ⋂ a∈In Va. Consider U = ⋃ n∈N � ⋃ a∈In Ua � . We have that U is an open set in X and A⊂ U , because for each y ∈ A by (ii) ∃n0 ∈ N such that ∀n ∈ N if n ≥ n0 there exists a ∈ In such that y ∈ Ua. Hence ⋃ a∈In Ua ⊂ ⋃ n∈N � ⋃ a∈In Ua � = U . Now we prove that V and U are disjoint sets. Suppose that there exists y ∈ V ∩ U . Then y ∈ U = ⋃ n∈N � ⋃ a∈In Ua � hence ∃n0 ∈ N such that y ∈ ⋃ a∈In Ua then ∃a0 ∈ In0 such that y ∈ Ua0 , but since y ∈ V and ∀n ∈ N V ⊂ ⋂ a∈In Va then we have that V ⊂ ⋂ a∈In0 Va, hence y ∈ V ⊂ ⋂ a∈In0 Va then ∀a ∈ In0 y ∈ Va which implies that y ∈ Va0 and then Va0 ∩ Ua0 6= ;, contradiction. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 520 Proposition 8. Let 〈X , T 〉 be a Hausdorff C-space and A be a Hurewicz subspace of X , then A is closed. Proof. We are going to show that if x ∈ Ac there is U ∈ T such that x ∈ U ⊂ Ac . Since x /∈ A, A is Hurewicz and X is a Hausdorff C-space then by Lemma 1, there are V and U disjoint open sets containing x and A, respectively. Hence we have that x ∈ V ⊂ Ac , because since A⊂ U and U ∩ V = ; then A∩ V = ;. So Ac is open, hence A is closed. Proposition 9. Let 〈X , TX 〉 be a topological C-space and let 〈Y, TY 〉 be a subspace of X , then Y is a C-space. Proof. Let x ∈ Y , consider {An}n∈N a sequence such that for each n ∈ N, An = {An j ∈ TY ; j = 1 . . . , kn} and x ∈ kn ⋂ j=1 An j then, since Y is subspace of X , ∀n ∈ N and ∀ j = 1, . . . , kn, there exists Un j ∈ TX such that An j = Un j ∩ Y . Hence An j ⊂ Un j , then ∀n ∈ N, x ∈ kn ⋂ j=1 Un j . Since X is a C-space, there exists U ∈ TX such that for each n ∈ N, x ∈ U ⊂ kn ⋂ j=1 Un j , hence V = U ∩ Y , V ∈ TY , x ∈ V and for each n ∈ N V = U ∩ Y ⊂ kn ⋂ j=1 Un j ! ∩ Y = kn ⋂ j=1 (Un j ∩ Y ) = kn ⋂ j=1 An j . So Y is a C-space. Proposition 10. Let 〈X , TX 〉 be a topological C-space, 〈Y, TY 〉 be a topological space and f : X → Y be an open, continuous and surjective function. Then Y is a C-space. Proof. Let y ∈ Y and {An}n∈N a sequence such that for each n ∈ N, An = {An j ∈ TY ; j = 1 . . . , kn} and y ∈ kn ⋂ j=1 An j . Since f is a surjection, ∃x ∈ X such that f (x) = y . For each n ∈ N and for each j = 1, . . . , kn, since f is continuous and An j ∈ TY , we have f −1(An j) ∈ TX . Hence for each n ∈ N, x ∈ f −1({y}) ⊂ f −1( kn ⋂ j=1 An j) = kn ⋂ j=1 f −1(An j). Because X is a C-space, there exists U ∈ TX such that for each n ∈ N, x ∈ U ⊂ kn ⋂ j=1 f −1(An j), then for each n ∈ N y = f (x) ∈ f (U) ⊂ f ( kn ⋂ j=1 f −1(An j)) ⊂ kn ⋂ j=1 An j , since f is an open function we have that f (U) is an open set. So Y is a C-space. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 521 4. Locally Hurewicz Spaces Definition 8. A topological space 〈X , T 〉 is locally Hurewicz if and only if, for each x ∈ X and for each V ∈ T with x ∈ V , there exists U ∈ T and H Hurewicz, where x ∈ U ⊂ H ⊂ V . Definition 9. A topological space 〈X , T 〉 is weakly locally Hurewicz if and only if, for each x ∈ X , there exists U ∈ T and H Hurewicz, such that x ∈ U. Definition 10. A topological space 〈X , T 〉 is relatively locally Hurewicz if and only if, for each x ∈ X , there exists U ∈ T with U Hurewicz, such that x ∈ U. Proposition 11. If 〈X , T 〉 is a Hurewicz topological space then 〈X , T 〉 is weakly locally Hurewicz. Proof. For each x ∈ X , consider U = X and H = X , then U ∈ T , H is Hurewicz and x ∈ U ⊂ H. Hence X is weakly locally Hurewicz. Proposition 12. If 〈X , T 〉 is a Hurewicz topological space then 〈X , T 〉 is relatively locally Hurewicz. Proof. For each x ∈ X , consider U = X . Then x ∈ U and U = X , hence U is Hurewicz. Therefore X is relatively locally Hurewicz. Proposition 13. If 〈X , T 〉 is a locally Hurewicz topological space then 〈X , T 〉 is weakly locally Hurewicz. Proof. For each x ∈ X , since X is locally Hurewicz, for V = X there are U open set in X and H Hurewicz such that x ∈ U ⊂ H ⊂ X . Hence X is a weakly locally Hurewicz space. Proposition 14. If 〈X , T 〉 is a relatively locally Hurewicz topological space then 〈X , T 〉 is weakly locally Hurewicz. Proof. For each x ∈ X , because X is locally Hurewicz, there exists U open set in X , with U Hurewicz such that x ∈ U . Then x ∈ U ⊂ U with U Hurewicz. Hence X is a weakly locally Hurewicz space. Proposition 15. A Hausdorff C-space 〈X , T 〉 is locally Hurewicz if and only if, X is weakly locally Hurewicz. Proof. (⇒) Proposition 13. (⇐) Consider x ∈ X and V a neighborhood of x . Since X is weakly locally Hurewicz there exist U ∈ T and H Hurewicz with x ∈ U ⊂ H. Consider A = H ∩ V c , since x ∈ V we have that x /∈ A and A is Hurewicz because H is Hurewicz and X is a Hausdorff C-space by Proposition 8 H is closed, then A = H ∩ V c , is a closed set. Since A ⊂ H by Proposition 6 A is Hurewicz, then by Lemma 1 there are Wx and WA disjoint open sets containing x and A, respectively. Consider U =Wx ∩ int(H). Then U is an open set containing x and U ⊂ int(H) ⊂ H, then U ⊂ H = H. Therefore U is a closed subspace of a Hurewicz space hence by Proposition 6 U is Hurewicz. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 522 Now we prove, that U ⊂ V . We have U ∩ A = ;. In fact, supposing by contradiction that ∃y ∈ U ∩ A, then y ∈ U ∩ A, then y ∈ U hence any neighborhood of y intersects U and y ∈ A ⊂ WA then WA ∩ U 6= ;, but U ⊂ Wx he have that WA ∩Wx 6= ;, contradiction. Hence U ⊂ H and U ∩ A= ;, which implies that U ⊂ V , then we have that x ∈ U ⊂ U ⊂ V . Therefore, X is a locally Hurewicz space. Proposition 16. A Hausdorff C-space 〈X , T 〉 is relatively locally Hurewicz if and only if, X is weakly locally Hurewicz. Proof. (⇒) Proposition 14. (⇐) Consider x ∈ X . Because X is weakly locally Hurewicz, there are U ∈ T and H Hurewicz such that x ∈ U ⊂ H. So U ⊂ H and since H is a Hurewicz subspace of a Hausdorff C-space by Proposition 8 H is closed. Then U ⊂ H, but U being closed by Proposition 6 U is Hurewicz, then x ∈ U , with U Hurewicz. So X is relatively locally Hurewicz space. Proposition 17. If 〈X , T 〉 is a locally compact topological space then 〈X , T 〉 is locally Hurewicz. Proof. For each x ∈ X and V a neighborhood of x , since X is locally compact there are U ∈ T and a compact set C with x ∈ U ⊂ C ⊂ V , but by Corollary 1, C is Hurewicz then X is locally Hurewicz. Proposition 18. If 〈X , T 〉 is a weakly locally compact topological space then 〈X , T 〉 is weakly locally Hurewicz. Proof. Consider x ∈ X . Because X is weakly locally compact there are U ∈ T and a compact set C such that x ∈ U ⊂ C . Since by Corollary 1 C is Hurewicz we have X weakly locally Hurewicz. Proposition 19. If 〈X , T 〉 is a relatively locally compact topological space then 〈X , T 〉 is relatively locally Hurewicz. Proof. Consider x ∈ X . Because X is relatively locally compact there is U ∈ T with U compact with x ∈ U . By Corollary 1, U is Hurewicz. Therefore X relatively locally Hurewicz. Example 5. Consider R in its topology, we have that R is Hurewicz, then by Proposition 11 and by Proposition 12 we have that R is weakly locally Hurewicz and relatively locally Hurewicz. We have that R is not a C-space, but it is a locally Hurewicz space because for each x ∈ R and V open in R, there exists ǫ > 0 such that x ∈ (x − ǫ, x + ǫ) ⊂ V , but (x − ǫ2 , x + ǫ2) is such that x ∈ (x − ǫ2 , x + ǫ2) ⊂ [x − ǫ 2 , x + ǫ2] ⊂ (x − ǫ, x + ǫ) ⊂ V , where (x − ǫ2 , x + ǫ2) is an open set and [x − ǫ2 , x + ǫ2] is compact, hence by Corollary 1 is Hurewicz. C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 523 Example 6. Consider R with the discrete topology. We have that R is not Lindelöf, because {{x}/x ∈ R} is an open covering of R which does not have a countable subcovering, hence by Proposition 3 R with the discrete topology is not Hurewicz, but for each x ∈ R, {x} is a compact set, then {x} is Hurewicz. Then we have that R with the discrete topology is locally Hurewicz, because for each x ∈ R and a neighborhood V of x, we have that x ∈ {x} ⊂ V , where {x} is an open set and Hurewicz. Since R with the discrete topological is a Hausdorff C-space, we have that R is weakly locally Hurewicz and relatively locally Hurewicz. Example 7. Consider R with the particular point p topology (Definition 4), by Example 2 we have that R is not Hurewicz. Given a nonempty open set A in R, we have that A = R, because for y ∈ R, we have that any neighborhood of y being nonempty contains p, then intersects A. Hence R is not relatively locally Hurewicz, because for each x ∈ R any neighborhood V of x is a nonempty set, then V = R, but since R is not Hurewicz, then we can not obtain a neighborhood of x where the closure is Hurewicz. But R is weakly locally compact, because for each x ∈ R we have that x ∈ {x , p} and {x , p} is an open and compact set, hence by Proposition 18, R is weakly locally Hurewicz. Proposition 20. Let 〈X , TX 〉 and 〈Y, TY 〉 be topological spaces, where X is locally Hurewicz and let f : X → Y be a continuous, open and surjective function, then Y is locally Hurewicz. Proof. Consider y ∈ Y and V a neighborhood of y , then ∃x ∈ X such that f (x) = y . By the continuity of f , we have that f −1(V ) ∈ TX . Because X is locally Hurewicz, there are U open set in X and H Hurewicz such that x ∈ U ⊂ H ⊂ f −1(V ), then y = f (x) ∈ f (U) ⊂ f (H) ⊂ f ( f −1(V )) ⊂ V, since f is an open function f (U) is open in Y and since f is continuous f (H) is Hurewicz by Proposition 4. So Y is a locally Hurewicz space. Proposition 21. Let 〈X , TX 〉 and 〈Y, TY 〉 be topological spaces, where X is weakly locally Hurewicz and let f : X → Y be a continuous, open and surjective function, then Y is weakly locally Hurewicz. Proof. By consider y ∈ Y , ∃x ∈ X such that f (x) = y . Because X is weakly locally Hurewicz, there are U ∈ TX and H Hurewicz with x ∈ U ⊂ H, then y = f (x) ∈ f (U) ⊂ f (H), since f is an open we have that f (U) ∈ TY and by the continuity of f and Proposition 4 f (H) is Hurewicz. So Y is a weakly locally Hurewicz space. Proposition 22. Let 〈X , TX 〉 and 〈Y, TY 〉 be topological spaces, with X is relatively locally Hurewicz and Y a Hausdorff C-space and let f : X → Y be a continuous, open surjection, then Y is relatively locally Hurewicz. Proof. Consider y ∈ Y , then ∃x ∈ X with f (x) = y . From the fact that X is relatively locally Hurewicz, there is U ∈ TX , with U Hurewicz with x ∈ U , then y = f (x) ∈ f (U) and C. Roika, S. Kudri, T. Breuckmann / Eur. J. Pure Appl. Math, 8 (2015), 514-525 524 since U Hurewicz and f is continuous by Proposition 4 f (U) is Hurewicz, but f (U) ⊂ f (U). Since Y is a Hausdorff C-space and f (U) is Hurewicz, we have that f (U) is closed, then f (U) ⊂ f (U), then f (U) is a closed subspace of a Hausdorff space, then f (U) is Hurewicz, so y ∈ f (U) ⊂ f (U), with f (U) is open and f (U) Hurewicz. So Y is a relatively locally Hurewicz space. Proposition 23. Let 〈X , TX 〉 be a locally Hurewicz topological space and let 〈Y, TY 〉 be a closed subspace of X , then Y is locally Hurewicz. Proof. Consider y ∈ Y and V an open set in Y such that y ∈ V . Then y ∈ X and there is V ′ opens in X , such that V = V ′ ∩ X . Since X is locally Hurewicz, there are U ′ open in X and H Hurewicz such that y ∈ U ′ ⊂ H ⊂ V ′, but since y ∈ Y we have that y ∈ U ′ ∩ Y ⊂ H ∩ Y ⊂ V ′ ∩ Y = V , where U ′ ∩ Y is open in Y and H ∩ Y is Hurewicz by Proposition 7. Therefore Y is locally Hurewicz. Proposition 24. Let 〈X , TX 〉 be a weakly locally Hurewicz topological space and let 〈Y, TY 〉 be a closed subspace of X , then Y is weakly locally Hurewicz. Proof. Consider y ∈ Y . Then there is U ∈ TX and H Hurewicz in X such that y ∈ U ⊂ H. So we have that y ∈ U ∩ Y ⊂ H ∩ Y , where U ∩ Y is open in Y and H ∩ Y is Hurewicz by Proposition 7, then Y is weakly locally Hurewicz. Proposition 25. Let 〈X , TX 〉 be a relatively locally Hurewicz topological space and let 〈Y, TY 〉 be a closed subspace of X , then Y is relatively locally Hurewicz. Proof. Consider y ∈ Y . Then there exists U ∈ TX with U Hurewicz and y ∈ U . Then y ∈ U∩Y and by Proposition 7 U∩Y is Hurewicz, but U ∩ Y ⊂ U∩Y = U∩Y . By Proposition 6 we have that U ∩ Y is Hurewicz, hence Y is relatively locally Hurewicz. Proposition 26. Let X be a weakly locally Hurewicz topological space then A⊂ X is open in X if and only if, A∩ H is open in H for each H Hurewicz. Proof. (⇒) If A is open in X then A∩ H is open in H. (⇐) Consider a ∈ A. We have that a ∈ X and since X is weakly locally Hurewicz there are U ∈ TX and H Hurewicz in X such that a ∈ U ⊂ H. Since H is Hurewicz, A∩ H is open in H, then since A∩U = (A∩H)∩U we have that A∩U is open in U , then there is V ∈ TX such that A∩ U = U ∩ V , but since U and V are open in X we have that U ∩ V is open in X and U ∩ A is open in X , hence a ∈ A∩ U ⊂ A, whit A∩ U open in X . Then A is open. Proposition 27. Let X be a locally Hurewicz topological space, then a subset A of X is open in X if and only if, A∩ H is open in H for each H Hurewicz. 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