ITES2001 Fourth Italian-Spanish conference on General Topology and its applications

Bressanone, 27-30 June 2001

Dense embeddings in ( pathwise ) connected spaces : results and open problems

Attilio Le Donne Università di Roma ≪La Sapienza≫, Italy

A space X is called ( pathwise ) connectifiable if it can be embedded as a dense subspace of a ( pathwise ) connected Hausdorff space Y; if such is the case, Y is called a ( pathwise ) connectification of X.

In this lecture we provide a survey of the most important results obtained by several authors. Moreover, we will give a rather complete endup-to-date list of the major open problems in this field.

References

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[2] O. T. Alas, M. G. Tkačenko, V. V. Tkachuk, R. G. Wilson, Connectedness and local connectedness of topological groups and extensions, Comment. Math. Univ. Carolin. 40 (1999), no. 4, 735–753.

[3] C. Costantini, A. Fedeli, A. Le Donne, Filters and pathwise connectifications , Rend. Istit. Mat. Univ. Trieste , XXXII (2001), 1–15.

[4] A. Emeryk, W. Kulpa, The Sorgenfrey line has no connected compactification, Comment. Math. Univ. Carolinae 18 (1977), no. 3, 483–487.

[5] A. Fedeli, A. Le Donne, One-point connectifications of subspaces of the Euclidean line, Rend. Mat. Appl. (7) 18 (1998), no. 4, 677–682 (1999). 54D05 (54C25)

[6] A. Fedeli, A. Le Donne, Dense embeddings in pathwise connected spaces, Topology Appl. 96 (1999), no. 1, 15–22. 54D05 (54C25)

[7] A. Fedeli, A. Le Donne, On locally connected connectifications, Topology Appl. 96 (1999), no. 1, 85–88. 54D05 (54C25)

[8] A. Fedeli, A. Le Donne, An independency result in connectification theory, Comment. Math. Univ. Carolin. 40 (1999), no. 2, 331–334. 54D05 (03E35 54C25 54D25)

[9] A. Fedeli, A. Le Donne, Connectifications and open components, Questions Answers Gen. Topology 18 (2000), no. 1, 41–45.

[10] A. Fedeli, A. Le Donne, On subconnected spaces, Questions Answers Gen. Topology 18 (2000), no. 1, 97–102.

[11] A. Fedeli, A. Le Donne, The Sorgenfrey line has a locally pathwise connected connectification, Proc. Amer. Math. Soc. 129 (2001), no. 1, 311–314. 54D05 (54D35)

[12] A. Fedeli, A. Le Donne, ω-connectifications and product spaces, Questions Answers Gen. Topology 18 (2000), no. 2, 283–288.

[13] G. Gruenhage, J. Kulesza, A. Le Donne, Connectifications of metrizable spaces, Special volume in memory of Kiiti Morita. Topology Appl. 82 (1998), no. 1-3, 171–179.

[14] J. r. Porter, R. G. Woods, Subspaces of connected spaces, Topology Appl. 68 (1996), no. 2, 113–131.

[15] S. Watson, R. G. Wilson, Embeddings in connected spaces, Houston J. Math. 19 (1993), no. 3, 469–481.