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Weakly Quotient Map and Space

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Weakly Quotient Map and Space Special Issue - 2020 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 RTICCT - 2020 Conference Proceedings Weakly Quotient Map and Space Chidanand Badiger T Venkatesh Department of Mathematics, Department of Mathematics, Rani Channamma University, Belagavi-591 156, Rani Channamma University, Belagavi-591 156, Karnataka, INDIA. Karnataka, INDIA. Abstract- In this paper, we introduce the new weakly quotient Definition 2.1 [9] A subset F of a space M is said to be map (briefly, 푤-quotient map), strongly 푤-quotient map and Weakly closed (briefly, w-closed) if cl(F) ⊆ U whenever 푤*-quotient map. Here will discuss composition of few such F ⊆ U and U is semi-open in M. Respectively called w-open quotient maps. Also, we investigate some important properties set, if M ∖ F is w-closed set in M. We denote the set of all w- and consequences associated with usual quotient and bi, tri- quotient maps. open sets in M by wO(M). Definition 2.2 A map h: M → N is said to be 푤-continuous −1 Key Words: 푤-quotient map, strongly 푤-quotient map, 푤*-quotient map, if h (F) is 푤-closed set of M, for every closed set F of map, 푤-quotient space. N. AMS Classification: 54A05, 54C10, 54H99. Definition 2.3 A map h: M → N is said to be 푤-irresolute map, if h−1(F) is 푤-closed set of M, for every 푤-closed set F I. INTRODUCTION of N. Construction is needed in any branch of science, Definition 2.4 A bijection h: M → N is called 푤- which can be sufficiently achieved by the theory of quotient homeomorphism, if both h and h−1 are 푤-continuous. notion in Mathematics. We can see how identifications Definition 2.5 (Consider) A topological space M is called explain or yield simple objects like circle, sphere, suspension, ηw-space, if every 푤-closed set is closed set. and torus. In fact not only special objects, most of the objects Theorem 2.6 Every 푤-irresolute maps are 푤-continuous. are actually quotient creations. Allen Hetcher [1] has III. 푤-QUOTIENT MAP discussed mapping cylinder, mapping cone etc. This takes We introduce here some class of maps 푤-quotient map, frontiers part of abstract quotient construction. Mobius strips, strongly 푤-quotient map, 푤*-quotient map in topological Klein bottle are quotient spaces, which have classical space. Investigate some relations between them with usual importance having special topological properties. quotient maps. Also we characterize few notions. Visualization is new aspect in the theory of gluing, but many Definition 3.1 Let M and N be two topological spaces then a time we unable to catch the object. surjective map h: M → N is said to be 푤-quotient map if h is There are many situations in topology where we build a 푤-continuous and h−1(V) is open in M implies V is 푤-open topological space by starting with some set (space) and doing set in N. some kind of “gluing” or “identifications” that makes suitable Example 3.2 Let h: ( ℝ, η ) → ( ℝ, η ) and η be usual object for our claim. In the literature of Mathematics, u u u topology on ℝ , the map h is defined by h(x) = 5x is quotient maps are generally called strong continuous maps or obviously (See 3.3) 푤-quotient map. identification maps, because of strong conditions of Theorem 3.3 Every quotient map is 푤-quotient map. continuity i.e. h−1(V) is open in M if and only if V is open in Proof: Let h: M → N be any quotient map, by definition h is N of a surjective fiunction h: M → N and their importance for surjective. It is known that every continuous map is 푤- the philosophy of gluing. [9] In 2000, M.Sheik John, introduced and studied w-closed sets respective properties continuous map, hence h is 푤-continuous. Since h is a −1( ) map in topological space. Lellis Thivagar [7], Chidanand quotient map, h V is open in M, implies V is open in N. Badiger et al. [2] and Balamani et al. [10] discussed As every open set is 푤-open set, implies V is 푤-open set in N. generalized quotient maps, rw-quotient map and ψ∗α - Therefore h is 푤-quotient map. quotient maps in topological space respectively. Bi-quotient, Theorem 3.4 [5; § 22] If h: M → N is 푤-continuous and Tri-quotient maps studied by E. Michal [4,3]. k: N → M is continuous, such that h ∘ k: N → N is identity We introduce such class of maps called 푤-quotient map, then h is 푤-quotient map. strongly 푤-quotient map, 푤*-quotient map in topological Proof: Since h ∘ k = Id implies h is bijective. h is 푤- −1 space. Which have many rich consequences concern to usual continuous by hypothesis. For any V ⊂ N with h (V) be −1 −1 quotient and bi, tri-quotient maps. Here we have given open in M , continuity of k gives the k (h (V)) = example and counter examples for respective results. (h ∘ k)−1(V) = Id−1(V) = V is open in N, implies V is 푤- open set in N. Therefore h is 푤-quotient map. II. PRELIMINARIES Theorem 3.5 h become 푤-quotient map, whenever h: M → N Throughout this paper L, M, N, R and S represent the is surjective, continuous and open map. topological spaces on which no separation axioms are Proof: By theorem 3.3, h is 푤-quotient map, because every assumed unless otherwise mentioned. Map here mean h: M → N is surjective, continuous and open maps are function and for a subset F of topological space M, the M\F quotient map. denotes the complement of F in M. We recall the following Theorem 3.6 h become 푤-quotient map, whenever h: M → N definitions. is surjective, continuous and closed map. Volume 8, Issue 12 Published by, www.ijert.org 62 Special Issue - 2020 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 RTICCT - 2020 Conference Proceedings Theorem 3.7 h become 푤-quotient map, whenever h: M → N Remark 5.4 Composition of two 푤-quotient maps need not is surjective, 푤-continuous and 푤-open map. be 푤-quotient map. Theorem 3.8 h become 푤-quotient map, whenever h: M → N Theorem 5.5 If N is ηw topological space, h: M → N is 푤- is surjective, 푤-continuous and 푤-closed map. quotient map and k: N → R is quotient map then k ∘ h is 푤- Proof: Two conditions are obvious by the hypothesis. For last quotient map. condition, any V ⊂ N with h−1(V) be open in M, implies M ∖ Proof: Obviously k ∘ h is surjective and for every open set h−1(V) is closed set in M. Since h is 푤-closed map implies U ⊂ R, the (k ∘ h)−1(U) = h−1(k−1(U)) which is 푤-open set h(M ∖ h−1(V)) = N ∖ V is 푤-closed in N. Therefore V is 푤- in M, implies k ∘ h is 푤-continuous. For every U ⊂ R with open set in N, h is 푤-quotient map. (k ∘ h)−1(U) = h−1(k−1(U)) is open in M, implies k−1(U) is IV. STRONGLY 푤-QUOTIENT AND 푤∗-QUOTIENT 푤-open set in N, by the hypothesis k is quotient map. Hence MAP U is open set as well 푤-open set in R. Definition 4.1 Let M and N be two topological spaces, a Theorem 5.6 If N is ηw topological space, h: M → N is surjective map h: M → N is said to be strongly 푤-quotient quotient map and k: N → R is 푤-quotient map then k ∘ h is 푤- map provided V ⊂ N is open in N if and only if h−1(V) is 푤- quotient map. open in M. Proof: Similar arguments with essential changes in 5.5 can Definition 4.2 Let M and N be two topological spaces, a works. surjective map h: M → N is said to be 푤*-quotient map if h is Corollary 5.7 If N is ηw topological space, h: M → N is 푤- 푤-irresolute and h−1(V) is 푤-open in M implies V is open set quotient map and k: N → R is 푤-quotient map then k ∘ h is 푤- in N. quotient map. Theorem 4.3 Every injective 푤*-quotient map is 푤*-open Proof: It fallows by combining the theorems 5.5 and 5.6 and map. (Converse need not hold). Proof: Let h: M → N be any injective 푤*-quotient map, For Theorem 5.8 If h: M → N is a strong 푤–quotient map and every 푤-open set V in M, injective gives h−1(h(V)) = V is k: N → R is a quotient map then koh is strong 푤-quotient 푤-open set in M. Since h is 푤*-quotient map, implies h(V) is map. open set as well 푤-open in N. Proof: Here surjective of k and h gives koh is surjective and Theorem 4.4 Every injective 푤*-quotient map is 푤*-closed koh is 푤–continuous map due to composition of continuous map. and 푤- continuous. Lastly if h−1(k−1(V)) is open in M. Proof: Let h: M → N be injective 푤*-quotient map, For every Since h is strong 푤-quotient map and quotient of k implies V 푤-closed set F in M , implies M ∖ F is 푤-open set in M .
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