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O.T. Alas, M.G. Tkaˇcenko, V.V. Tkachuk, R.G. Wilson +A?JA@AII =@ ?= ?A?JA@AII B JFCE?= CHKFI =@ ANJAIEI

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O.T. Alas, M.G. Tkaˇ cenko, V.V. Tkachuk, R.G. Wilson Connectedness and local connectedness of topological groups and extensions

Comment.Math.Univ.Carolinae 40,4 (1999) 735-753.

Abstract: It is shown that both the free topological group F(X) and the free Abelian topological group A(X) on a connected locally connected spaceX are lo- cally connected. For the Graev’s modification of the groupsF(X) andA(X), the corresponding result is more symmetric: the groups FΓ(X) and AΓ(X) are con- nected and locally connected ifX is. However, the free (Abelian) totally bounded groupF T B(X) (resp.,AT B(X)) is not locally connected no matter how “good” a spaceX is.

The above results imply that every non-trivial continuous homomorphism ofA(X) to the additive group of reals, withX connected and locally connected, is open.

We also prove that any dense in itself subspace of the Sorgenfrey line has a Urysohn connectification. IfD is a dense subset of{0,1}cof power less thanc, thenD has a Urysohn connectification of the same cardinality asD.

We also strengthen a result of [1] for second countable Tychonoff spaces without open compact subspaces proving that it is possible to find a compact metrizable connectification of such a space preserving its dimension if it is positive.

Keywords: connected, locally connected, free topological group, Pontryagin’s du- ality, pseudo-open mapping, open mapping, Urysohn space, connectification AMS Subject Classification: Primary 54H11, 54C10, 22A05, 54D06; Secondary 54D25, 54C25

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