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A.V. Arhangel’skii, R.Z. Buzyakova +LAHCA?A E ?F=?J= =@ EA=H E@ABAII

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A.V. Arhangel’skii, R.Z. Buzyakova

Convergence in compacta and linear Lindelofness

Comment.Math.Univ.Carolinae 39,1 (1998) 159-166.

Abstract: LetX be a compact Hausdorff space with a pointxsuch thatX\ {x}

is linearly Lindel¨of. Is thenX first countable atx? What if this is true for everyx in X? We consider these and some related questions, and obtain partial answers;

in particular, we prove that the answer to the second question is “yes” when X is, in addition,ω-monolithic. We also prove that if X is compact, Hausdorff, and X \ {x} is strongly discretely Lindel¨of, for every x in X, then X is first count- able. An example of linearly Lindel¨of hereditarily realcompact non-Lindel¨of space is constructed. Some intriguing open problems are formulated.

Keywords: point of complete accumulation, linearly Lindel¨of space, local com- pactness, first countability,κ-accessible diagonal

AMS Subject Classification: 54F99, 54D30, 54E35

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