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Preservation of a Convergence of a Sequence to a Set Akira Iwasa University of South Carolina - Beaufort, [email protected]

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Preservation of a Convergence of a Sequence to a Set Akira Iwasa University of South Carolina - Beaufort, Iwasa@Uscb.Edu University of South Carolina Scholar Commons Department of Mathematics and Computational Faculty Publications Science 2014 Preservation of a Convergence of a Sequence to a Set Akira Iwasa University of South Carolina - Beaufort, [email protected] Masaru Kada Osaka Prefectural University Shizuo Kamo Osaka Prefectural University Follow this and additional works at: https://scholarcommons.sc.edu/ beaufort_math_compscience_facpub Part of the Applied Mathematics Commons Publication Info Preprint version Topology Proceedings, Volume 44, 2014, pages 97-105. © Topology Proceedings 2014, Auburn University This Article is brought to you by the Department of Mathematics and Computational Science at Scholar Commons. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. PRESERVATION OF CONVERGENCE OF A SEQUENCE TO A SET AKIRA IWASA, MASARU KADA, AND SHIZUO KAMO Abstract. We say that a sequence of points converges to a set if every open set containing the set contains all but finitely many terms of the sequence. We investigate preservation of convergence of a sequence to a set in forcing extensions. 1. Introduction P Let hX; τi be a topological space and P a notion of forcing. LetSV denote the forcing extension of V by P. In VP, we define a topology τ P = f S : S ⊆ τg on X, that is, τ P is the topology generated by τ in VP. We say that a topological property ' is preserved by P if whenever hX; τi satisfies ', hX; τ Pi also satisfies '. First, let us observe the following: Theorem 1.1. Convergence of a sequence (to a point) is preserved by any forcing. P Proof. If a sequence fxn : n 2 !g in hX; τi converges to y, then in hX; τ i it still converges to y because in VP τ serves as a base for τ P. By the above theorem, there is nothing to investigate about preservation of convergence of a sequence. So let us generalize the concept of convergence. Definition 1.2. We say that a sequence of points fxn : n 2 !g converges to a set A if xn 2= A for all n 2 ! and for every open set U containing A, there exists k 2 ! such that for every n ≥ k, xn 2 U. Let us illustrate the fact that convergence of a sequence to a set is not necessarily preserved by forcing. Example 1.3. There exists a space X and a sequence fxn : n 2 Ng in X such that: (1) in V, the sequence fxn : n 2 Ng converges to a set A, and P (2) in V for some forcing P, fxn : n 2 Ng does not converge to A. Proof. Let [0; 1]V be the unit interval in V equipped with the usual topology and V fqn : n 2 Ng enumerate the rationals in [0; 1] . In the example, • X = [0; 1]V × [0; 1]V equipped with the usual topology. • 1 2 N xn = (qn; n ) for each n . • A = [0; 1]V × f0g. 2010 Mathematics Subject Classification. 54A20, 03E40. Key words and phrases. sequence, convergence, preservation, forcing. The second author was supported by Grant-in-Aid for Young Scientists (B) 21740080, MEXT. 1 2 AKIRA IWASA, MASARU KADA, AND SHIZUO KAMO • P is any forcing notion which adjoins a new real. f 1 2 Ng V×f g The sequence (qn; n ): n converges to the set [0; 1] 0 . We shall show that P f 1 2 Ng V ×f g in V , the sequence (qn; n ): n does not converge to the set [0; 1] 0 . Let P f 2 Ng r be a new real adjoined by such that 0 < r < 1. Take subsequences qni : i f 2 Ng and qmi : i of rationals converging to r such that ··· ··· qn1 < qn2 < qn3 < r < qm3 < qm2 < qm1 : Consider the following points: 1 1 1 1 (0; 1); (qn ; ); (qn ; ); ··· (r; 0) ··· ; (qm ; ); (qm ; ); (1; 1) 1 2n1 2 2n2 2 2m2 1 2m1 These points form a V-shape with (r; 0) being the tip of the letter V, and the tip (r; 0) is not in [0; 1]V ×f0g. Using the V-shape, we can define an open set containing V 1 1 the set [0; 1] × f0g and missing points (qn ; ) and (qm ; ) for all i 2 N. This i ni i mi P f 1 2 Ng implies that in V the sequence (qn; n ): n does not converge to the set [0; 1]V × f0g. In this note, we investigate under what circumstances convergence of a sequence to a set is preserved by forcing. Let us give definitions. Definition 1.4. We say that a sequence fxn : n 2 !g is of discrete points if for each k 2 !, xk 2= fxn : n 2 ! n fkgg. (In this paper, we only consider sequences of discrete points.) f g f 2 g f g f 2 g We write xn n for xn : n ! and xni i for xni : i ! . A Tychonoff space X is said to be pseudocompact if every continuous real-valued function on X is bounded. A point p is an accumulation point of a set A if for every neighborhood U of p, U \ A is infinite. A point p is a cluster point of a family fAn : n 2 !g if for every neighborhood U of p, U \ An =6 ; for infinitely many n 2 !. A space X is said to be perfect if for every x 2 X, x 2 X n fxg. F n(κ, 2) is the set of all finite partial functions from a cardinal κ to 2. ([8] Forcing with F n(κ, 2) adjoins κ-many Cohen reals.) A space X is said to be scattered if for every subspace S ⊆ X, there exists an x 2 S such that x2 = S n fxg. For a space X, X can be uniquely represented as X = P [S, where P is a perfect set, S is a scattered set and P \ S = ;; we say P is the perfect kernel of X and S the scattered kernel of X ([3] Problem 1.7.10). In this paper, we mainly deal with Tychonoff spaces. The property that a space is Tychonoff is preserved by any forcing ([2] Lemma 22). 2. Convergence of a sequence to a set In this section we study the concept of convergence of a sequence to a set. A proof of the following proposition is routine. Proposition 2.1. Let fxngn be a sequence of discrete points in a space X and let A be a subset of X such that fxngn \ A = ;. The following are equivalent: (1) fxngn converges to A. f g f g f g \ 6 ; (2) For every subsequence xni i of xn n, xni i A = . PRESERVATION OF CONVERGENCE OF A SEQUENCE TO A SET 3 We use the following characterization of pseudocompact spaces. Proposition 2.2. ([4] pp.177-178; [3] Theorem 3.10.23) For a Tychonoff space X, the following are equivalent: (1) X is pseudocompact. (2) If U = fUn : n 2 !g is a sequence of nonempty open subsets of X such that Ui \ Uj = ; whenever i =6 j, then U has a cluster point in X. ⊇ ⊇ · · · (3) ForT every decreasing sequence U1 U2 of non-empty open subsets of 6 ; X, n2! Un = . The following are equivalent conditions for a sequence to converge to a set. Proposition 2.3. Suppose that X is a Tychonoff space and that fxngn is a se- quence of discrete points in X. The following are equivalent: (1) fxngn converges to a set. (2) fxngn converges to the set fxngn n fxngn. f g f g f g (3) For every subsequence xni i of xn n, xni i has an accumulation point, f g n f g 6 ; that is, xni i xni i = . (4) The closure fxngn of fxngn is pseudocompact. Proof. (1) =) (2). Suppose that a sequence fxngn does not converge to fxngn n fxngn. Take an arbitrary set A ⊆ X such that A \ fxngn = ;. We shall show that the sequence fxngn does not converge to A. By Proposition 2.1, there exists a f g f g \ f g nf g ; f g ⊆ f g subsequence xni i such that xni i [ xn n xn n] = . Since xni i xn n, it f g ⊆ f g f g \ ; f g \ ; must be the case that xni i xn n. Since xn n A = , we have xni i A = . By Proposition 2.1, the sequence fxngn does not converges to A. ) f g f g f g \ (2) = (3). Assume on the contrary that xni i = xni i. Then xni i [fxngn n fxngn] = ;, which implies that fxngn does not converge to fxngn n fxngn by Proposition 2.1. (3) =) (4). Suppose Y := fxngn is not pseudocompact. According to Proposi- tion 2.2, there exists in Y a family of nonempty open sets U = fUi : i 2 !g such that Ui \ Uj = ; whenever i =6 j and U has no cluster point in Y . For each i 2 !, 2 f g pick xni Ui. xni i does not have an accumulation point in Y . Since Y is closed, f g xni i does not have an accumulation point in X either. (4) =) (1). It is clear that (2) implies (1), so it suffices to prove (4) implies (2). We assume on the contrary that fxngn does not converge to fxngn n fxngn. f g f g \ f g n By Proposition 2.1, there exists a subsequence xni i such that xni i [ xn n f g ; f g ⊆ f g f g xn n] = .
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