论文标题
关于拓扑空间及其$G_δ$修改的免费组数
On the free set number of topological spaces and their $G_δ$-modifications
论文作者
论文摘要
对于拓扑空间$ x $,我们建议将子集$ s \ subset x $“ free in $ x $”称为“ $ x $ in $ x $”,如果订购良好,将其变成$ x $的免费序列。然后,著名的红衣主教函数$ f(x)$可以定义为$ \ sup \ {| s | :s \ text {在} x \} $中是免费的,将称为$ x $的免费集数。 我们证明了涉及$ f(x)$和$ f(x_Δ)$的几种新不平等,其中$x_δ$是$g_Δ$ - $ x $的修改: $ \ bullet $ $ l(x)\ le 2^{2^{f(x)}} $如果$ x $是$ t_2 $和$ t_2 $和$ l(x)\ le 2^{f(x)} $,如果$ x $是$ t_3 $; $ \ bullet $ $ | x | \ le 2^{2^{f(x)\ cdotψ_c(x)}} \ le 2^{2^{f(x)\ cdotχ(x)}} $用于任何$ t_2 $ -space -space $ x $; $ \ bullet $ $ f(x_Δ)\ le 2^{2^{2^{f(x)}}} $如果$ x $是$ t_2 $ and $ t_2 $和$ f(x_Δ)\ le 2^{2^{f(x)}} $,如果$ x $是$ t_3 $。
For a topological space $X$ we propose to call a subset $S \subset X$ "free in $X$" if it admits a well-ordering that turns it into a free sequence in $X$. The well-known cardinal function $F(X)$ is then definable as $\sup\{|S| : S \text{ is free in } X\}$ and will be called the free set number of $X$. We prove several new inequalities involving $F(X)$ and $F(X_δ)$, where $X_δ$ is the $G_δ$-modification of $X$: $\bullet$ $L(X) \le 2^{2^{F(X)}}$ if $X$ is $T_2$ and $L(X)\le 2^{F(X)}$ if $X$ is $T_3$; $\bullet$ $|X|\le 2^{2^{F(X) \cdot ψ_c(X)}} \le 2^{2^{F(X) \cdot χ(X)}}$ for any $T_2$-space $X$; $\bullet$ $F(X_δ)\le 2^{2^{2^{F(X)}}}$ if $X$ is $T_2$ and $F(X_δ)\le 2^{2^{F(X)}}$ if $X$ is $T_3$.