A first countable linearly Lindelöf not Lindelöf topological space

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• 作者：Oleg Pavlov<sup>asup><sup> ; sup><sup>matematika.atiso@gmail.com"" rel=""nofollowsup>
• 关键词：Linearly Lindel&#xf6 ; f ; Lindel&#xf6 ; f ; First countable ; l6"">style=""text-decoration ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V1K-53DFY1C-2&_mathId=mml6&_pii=S016686411100304X&_issn=01668641&_acct=C000053662&_v
• 刊名：Topology and its Applications
• 出版年：2011
• 出版时间：1 October 2011
• 年：2011
• 卷：158
• 期：16
• 页码：2205-2209
• 全文大小：148 K

文摘

A topological space X is called linearly Lindel&#xf6;f if every increasing open cover of X has a countable subcover. It is well known that every Lindel&#xf6;f space is linearly Lindel&#xf6;f. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<<sub>ωsub>.

Arhangelskii and Buzyakova proved that the cardinality of a first countable linearly Lindel&#xf6;f space does not exceed 2<sup><sub>0sub>sup>. Consequently, a first countable linearly Lindel&#xf6;f space is Lindel&#xf6;f if <sub>ωsub>>2<sup><sub>0sub>sup>. They asked whether every linearly Lindel&#xf6;f first countable space is Lindel&#xf6;f in ZFC. This question is supported by the fact that all known linearly Lindel&#xf6;f not Lindel&#xf6;f spaces are of character at least <sub>ωsub>. We answer this question in the negative by constructing a counterexample from MA+<sub>ωsub><2<sup><sub>0sub>sup>.

A modification of Alsters Michael space that is first countable is presented.