Extensions of topological groups do not respect countable compactness Montserrat Bruguera Dept. de Matem´ atica Aplicada I Universidad Polit´ecnica de Catalu˜ na C/ Gregorio Mara˜ n´ on 44-50, 08028 Barcelona, Espa˜ na e-mail:
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Mikhail Tkachenko Departamento de Matem´ aticas Universidad Aut´ onoma Metropolitana Av. San Rafael Atlixco # 186, Col. Vicentina, Iztapalapa, C.P. 09340, Mexico D. F., Mexico
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Preliminary version of the article published in: Questions & Answers Gen. Topol. V. 22 no. 1 (2004), 33–37. Abstract We construct an Abelian topological group G and a closed subgroup N of G such that the groups N and G/N are countably compact, but G is not.
Let P be a (topological, algebraic, or topological-algebraic) property. We say that P is a three-space property if the following holds: whenever N is a 2000 Mathematics Subject Classification: Primary 54H11, 22A05; Secondary 54A20, 54G20. Key words and phrases: Topological group, countably compact group, ω-bounded group, sequentially complete
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closed normal subgroup of a topological group G and both N and G/N have P, the group G also has P. Compactness, completeness, precompactness, pseudocompactness, connectedness and metrizability are three-space properties in the class of topological groups (see [1, 8]). The problem of whether countable compactness is a three-space property, in the special case of products of two countably compact groups, was attacked by van Douwen [4], Hart and van Mill [7], Tomita [9], among others. However, all known constructions of counterexamples make use of extra set-theoretic assumptions like the Continuum Hypothesis or Martin’s Axiom. Here we present a short construction which shows that countable compactness strongly fails to be a three-space property in ZF C only. A topological group G is called sequentially complete if no sequence in G ˜ ˜ is the Ra˘ıkov completion of G. Clearly, converges to a point of G\G, where G countably compact groups are sequentially complete. A group G is said to be ω-bounded if every countable subset of G is contained in a compact subgroup. Obviously, every ω-bounded group is countably compact. It is easy to see that the classes of sequentially complete groups and ω-bounded groups are productive. The example below also shows that extensions of topological groups preserve neither sequential completeness nor ω-boundedness, thus answering Question 6.1 of [3] in the negative and stressing the difference between products and extensions in topological groups. Example 1 There exist an Abelian pseudocompact topological group G and a closed subgroup N of G such that N is ω-bounded, G/N is compact and metrizable, but G is not sequentially complete. In particular, the group G is not countably compact. Proof. Let c = 2ω . Consider the group Π = Z(2)c with the product topology and its dense subgroup N1 = {x ∈ Π : |supp(x)| ≤ ω}, where Z(2) = {0, 1} carries the discrete topology and supp(x) = {α < c : x(α) = 1} for each x ∈ Π . Choose x∗ ∈ Π \ N1 and put T = N1 + hx∗ i. Note that the group N1 is ω-bounded by [5, 3.10.D]. Clearly, T is also dense in Π . Let R = Z(2)ω . We construct an algebraic isomorphism ϕ: T → R as follows. Consider the group Π as a linear space over the field Z(2) and take a Hamel basis Z of N1 ; then |Z| = c and X = Z ∪ {x∗ } is a Hamel basis of T . For every n ∈ ω, choose an element hn ∈ Z such that hn 6= hm if n 6= m. Take a sequence {yn : n ∈ ω} of elements algebraically independent in R which converges to 0. Let Y be a Hamel basis of R such that {yn : n ∈ ω} ⊆ Y , and 2
choose a point y ∗ ∈ Y \ {yn : n ∈ ω}. It is clear that |Y | = |R| = c. We put ϕ(x∗ ) = y ∗ and ϕ(hn ) = y ∗ +yn , for each n ∈ ω. Extend ϕ to an injective map of X to R such that ϕ(X \ ({hn : n ∈ ω} ∪ {x∗ })) = Y \ ({yn : n ∈ ω} ∪ {y ∗ }). Finally, we extend ϕ to a homomorphism of T to R which is denoted by the same symbol ϕ. Note that if xn = x∗ +hn , then ϕ(xn ) = ϕ(x∗ )+ϕ(hn ) = y ∗ +(y ∗ +yn ) = yn . Hence Y ⊆ ϕ(T ). Since Y is a Hamel basis of R, we have ϕ(T ) = R. In addition, ϕ is a monomorphism. Indeed, if there exists t ∈ T \ {0} such that ϕ(t) = 0, we can write (reordering indices if necessary) t = h1 + . . . + hp + x∗ + s1 + . . . + sr with si ∈ X \ ({hn : n ∈ ω} ∪ {x∗ }) for i = 1, . . . , r. Then ϕ(t) = (y ∗ +y1 )+. . .+(y ∗ +yp )+y ∗ +ϕ(s1 )+. . .+ϕ(sr ) = (p + 1)y ∗ + y1 + . . . + yp + ϕ(s1 ) + . . . + ϕ(sr ) = 0, which is impossible because the set {y ∗ } ∪ {yn : n ∈ ω} ∪ {ϕ(si ) : i ∈ ω} ⊆ Y is lineally independent in R over Z(2). So, ϕ is an isomorphism. Let F = {(x, ϕ(x)) : x ∈ T } be the graph of ϕ. Then N = N1 × {0} and G = N + F are subgroups of Π × R and N = G ∩ (Π × {0}) is a closed ω-bounded subgroup of G. Since N is dense in Π × {0}, the quotient group G/N is topologically isomorphic to the compact metrizable group R by [6, Lemma 1.3]. Hence the group G is pseudocompact by [1, Theorem 6.3]. It remains to note that the group G = N + F is not sequentially complete because the sequence {(hn , 0) + (xn , ϕ(xn )) = (x∗ , yn ) : n ∈ ω} of elements in G converges to the point (x∗ , 0) which is not in G since x∗ 6∈ N1 . The group G in Example 1 is zero-dimensional being a subgroup of the group Z(2)c × Z(2)ω ∼ = Z(2)c . One can replace Z(2) by the circle group T in order to obtain a connected subgroup G0 of Tc × Tω ∼ = Tc and its closed subgroup N 0 with the properties as in Example 1. We mentioned above that extensions of topological groups preserve both compactness and metrizability. However, we do not know the answer to the following question: Problem 2 Let N be a closed normal subgroup of a topological group G and suppose that all compact sets in N and in G/N are metrizable. Is every compact subset of G metrizable?
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References [1] W.W. Comfort and L. Robertson, Extremal phenomena in certain classes of totally bounded groups, Dissert. Math. 272 (1988), 1–48. [2] W.W. Comfort and K.A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. [3] D. Dikranjan and M. Tkachenko, Sequentially complete groups: dimension and minimality, J. Pure Appl. Algebra 157 (2001), 215–239. [4] E.K. Douwen, van, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (1980), 417–427. [5] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989. [6] D.L. Grant, Topological groups which satisfy an open mapping theorem, Pacific J. Math. 68 (1977), 411–423. [7] K.P. Hart and J. van Mill, A countably compact group H such that H ×H is not countably compact, Trans. Amer. Math. Soc. 323 (1991), 811–821. [8] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, Die Grundlehrender Mathematischen Wissenschaften 115 (1963). [9] A.H. Tomita, A group under M Acountable whose square is countably compact but whose cube is not, Topology Appl. 91 (1999), 91–104.
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