-spaces

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• 作者：Taras Banakh ^{t.o.banakh@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54C35 ; 54E20 ; 22A30
• 刊名：Topology and its Applications
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：195
• 期：Complete
• 页码：151-173
• 全文大小：544 K

文摘

A regular topological space X   is defined to be a P₀${\mathfrak{P}}_0$-space   if it has countable Pytkeev network. A network N$\mathcal{N}$ for X is called a Pytkeev network   if for any point x∈X$x\in X$, neighborhood O_x⊂X$O_x\subset X$ of x   and subset 32afbd992a945d406e352038ef5f980" title="Click to view the MathML source">A⊂X$A\subset X$ accumulating at a x   there is a set N∈N$N\in \mathcal{N}$ such that N⊂O_x$N\subset O_x$ and N∩A$N\cap A$ is infinite. The class of P₀${\mathfrak{P}}_0$-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of ℵ₀${\mathrm{\aleph }}_0$-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an ℵ₀${\mathrm{\aleph }}_0$-space X   and a P₀${\mathfrak{P}}_0$-space Y   the function space C_k(X,Y)$C_k(X,Y)$ endowed with the compact-open topology is a P₀${\mathfrak{P}}_0$-space. For any sequential ℵ₀${\mathrm{\aleph }}_0$-space X   the free abelian topological group A(X)$A(X)$ and the free locally convex linear topological space a358c86eb9a506608" title="Click to view the MathML source">L(X)$L(X)$ both are P₀${\mathfrak{P}}_0$-spaces. A sequential space is a P₀${\mathfrak{P}}_0$-space if and only if it is an ℵ₀${\mathrm{\aleph }}_0$-space. A topological space is metrizable and separable if and only if it is a P₀${\mathfrak{P}}_0$-space with countable fan tightness.