A Hausdorff HS-space which is not regular
by
Attilio Le Donne
University of Roma

In this paper I disprove, with a counterexample, a theorem of H.-J. Schmidt [S], which states that each Hausdorff HS-space is regular. M. Paoli and E. Ripoli [PR1] noted that the proof of this theorem is incorrect, but they left the statement open.

A topological space X is called a HS-space if, for every subspace A of X the map i_A : C(A) --> C(X), defined by i_A(B) = cl_X (B), for each B in C(A), is a continuous map, where we denote with C(X) the set of all non-empty closed subsets of X, with the Tychonoff topology, which is generated by the sets C(X, U) = { F in C(X) : F subset U }, for each F open subset of X

In [S] H.-J. Schmidt gave it as a theorem that if a HS-space is Hausdorff then it is necessarily regular. M. Paoli and E. Ripoli in [PR1] and [PR2] noted that the proof of this theorem is incorrect, but the question of the correctness of the statement remained open.

S. Barov, G. Dimov and St. Nedev in [BDN1] [BDN2] gave a partial proof of the theorem, in particular they showed that the theorem of Schmidt is true for all spaces with countable character. About HS-space see also [Se].

In this paper I show, by constructing a counterexample, that the theorem of H.J. Schmidt is false.

Date received: July 22, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaj-51.