# zbMATH — the first resource for mathematics

Provisional solution to a Comfort–van Mill problem. (English) Zbl 1017.54022
Let $$\mathcal U$$ and $$\mathcal V$$ be two non-empty classes of topological groups and $$X$$ be a topological space. A topological group $$G=F(X, \mathcal U, \mathcal V)$$ is called a free $$(\mathcal U, \mathcal V)$$-group over $$X$$ if $$G\in\mathcal U$$, $$G$$ contains $$X$$ as a subspace, and every continuous map $$f$$ of $$X$$ to a group $$H\in\mathcal V$$ uniquely extends to a continuous homomorphism $$\widetilde f: G\to H$$. If such an extension $$\widetilde f: G\to H$$ is not required to be unique, then the group $$G=F_w(X, \mathcal U, \mathcal V)$$ is called a weakly free $$(\mathcal U, \mathcal V)$$-group over $$X$$. In the case $$\mathcal U=\mathcal V$$ these groups are called the free $$\mathcal U$$-group $$F(X, \mathcal U)$$ and the weakly free $$\mathcal U$$-group $$F_w(X, \mathcal U)$$ over $$X$$, respectively. W. W. Comfort and J. van Mill [Topology Appl. 29, 245-265 (1988; Zbl 0649.22001)] posed the following general problem: Does there exist a free $$(\mathcal U, \mathcal V)$$-group over $$X$$ for classes of topological groups $$\mathcal U$$ and $$\mathcal V$$ and a Tikhonov space $$X$$? The author proves that under Martin’s Axiom every Abelian group $$G$$ of non-measurable cardinality is the intersection of some family of countably compact subgroups of its Bohr compactification $$bG$$. Using this result he shows that weakly free countably compact topological groups do not exist. This gives an answer to the Comfort-van Mill problem. In fact, the author shows that under Martin’s Axiom a free $$(\mathcal P, \mathcal C\mathcal C)$$-group over a topological space $$X$$ exists if and only if $$X$$ is empty, where $$\mathcal P$$ and $$\mathcal C\mathcal C$$ are the classes of pseudocompact and countably compact topological groups, respectively. Nevertheless, he proves the existence of a weakly free $$(\mathcal P, \mathcal C\mathcal C)$$-group over a Tikhonov space $$X$$ and establishes that this construction is functorial. Also the author shows that similar results remain valid in the Abelian case. The paper closes with six open problems on this topic.
##### MSC:
 54H11 Topological groups (topological aspects) 54A35 Consistency and independence results in general topology 54B10 Product spaces in general topology 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54B05 Subspaces in general topology 22A05 Structure of general topological groups
Full Text:
##### References:
  Arhangel’skiı̌, A.V., Classes of topological groups, Russian math. surveys, 36, 151-174, (1981), Russian original in: Uspekhy Mat. Nauk 36 (1981) 127-146 · Zbl 0488.22001  Arhangel’skiı̌, A.V., On a theorem of W.W. comfort and K.A. ross, Comment. math. univ. carolin., 40, 133-151, (1999)  Comfort, W.W.; Grant, D.L., Cardinal invariants, pseudocompactness and minimality: some recent advances in the topological theory of topological groups, Topology proc., 6, 227-265, (1981) · Zbl 0525.22001  Comfort, W.W.; Hernandez, S.; Trigos-Arrieta, F.J., Relating a locally compact abelian group to its Bohr compactification, Adv. math., 120, 322-344, (1996) · Zbl 0863.22004  Comfort, W.W.; van Mill, J., On the existence of free topological groups, Topology appl., 29, 245-265, (1988) · Zbl 0649.22001  Comfort, W.W.; Remus, D., Compact groups of Ulam-measurable cardinality: partial converse to theorems of arhangel’skiı̆ and varopoulos, Math. japon., 39, 203-210, (1994) · Zbl 0817.22006  Comfort, W.W.; Ross, K.A., Topologies induced by groups of characters, Fund. math., 55, 283-291, (1964) · Zbl 0138.02905  Comfort, W.W.; Ross, K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. math., 16, 483-496, (1966) · Zbl 0214.28502  Comfort, W.W.; Saks, V., Countably compact groups and finest totally bounded topologies, Pacific J. math., 49, 33-44, (1973) · Zbl 0271.22001  Dikranjan, D.; Tkachenko, M., Sequential completeness of quotient groups, Bull. austral. math. soc., 61, 129-151, (2000) · Zbl 0943.22001  D. Dikranjan, M. Tkachenko, Varieties generated by countably compact Abelian groups, Proc. Amer. Math. Soc., to appear · Zbl 1017.22001  van Douwen, E.K., The product of two countably compact topological groups, Trans. amer. math. soc., 262, 417-427, (1980) · Zbl 0453.54006  van Douwen, E.K., The maximal totally bounded group topology on G and the biggest minimal G-space for abelian groups G, Topology appl., 34, 69-91, (1990) · Zbl 0696.22003  Flor, P., Zur Bohr-konvergenz von folgen, Math. scand., 23, 169-170, (1968) · Zbl 0182.36003  Fokkink, R., A note on pseudocompact groups, Proc. amer. math. soc., 107, 569-571, (1989) · Zbl 0683.22001  Hart, K.; van Mill, J., A countably compact group H such that H×H is not countably compact, Trans. amer. math. soc., 323, 811-821, (1991) · Zbl 0770.54037  Kakutani, S., Free topological groups and direct product of topological groups, Proc. imp. acad. Tokyo, 40, 595-598, (1944) · Zbl 0063.03105  Morris, S.A., Free abelian topological groups, (), 375-391  Ross, K.A.; Stromberg, K., Baire sets and Baire measures, Arch. math., 8, 151-160, (1965) · Zbl 0147.04501  Tkačenko, M.G., Compactness type properties in topological groups, Czech. math. J., 38, 324-341, (1988) · Zbl 0664.54006  Tkachenko, M.G., On subgroups of pseudocompact topological groups, Questions answers gen. topology, 10, 41-50, (1992) · Zbl 0748.22001  M.G. Tkachenko, Iv. Yaschenko, Independent group topologies on Abelian groups, Topology Appl., to appear · Zbl 0997.22003  Tomita, A.H., On finite powers of countably compact groups, Comment. math. univ. carolin., 37, 3, 617-626, (1996) · Zbl 0881.54022  Tomita, A.H., A group under MA_countable whose square is countably compact but whose cube is not, Topology appl., 91, 91-104, (1999) · Zbl 0927.54039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.