论文标题
相对（功能上）I型空间和狭窄子空间
Relative (functionally) Type I spaces and narrow subspaces
论文作者
论文摘要
一个开放式链条盖$ \ {u_α：α\inκ\} $（$κ$ a cardinal）的空间$ x $是系统的封面，如果$u_α$的关闭在$u_β$中包含在$u_β$中，而$α<β$，而$ x $ $ x $是type i，则是I type I type I If $κ= $κ=ω________________________________________________$u__α$是lincef。 $ x $的封闭子空间$ d \子集X $如果每个系统盖$ \ {v_α：α\inΩ_1\} $的$ x $，则$ d $的$ d $在$v_α$中包含在$v_α$中，或者与$v_α$ of $v_α$相交的每种$ d $ lindelouf的关闭。对于连续$ s：x \ to \ mathbb {l} _ {\ ge 0} $（其中$ \ mathbb {l} _ {l} _ {\ ge 0} $）定义IS在功能上狭窄的subspepaces和功能上。例如，$ \ mathbb {l} _ {\ ge 0} $，$ω_1$本身和任何其他空间都狭窄。 我们研究了这些属性和相对版本及其关系，并特别显示以下内容。在功能上，有功能上的I型空间在功能上不是I型I类型，而常规类型I空间是功能类型I的。我们展示了在某些但不在其他空间中的狭窄空间的示例。有一些tychonoff space $ y $的子空间在$ y $上缩小但不狭窄，而这两个概念都同意$ y $是否正常。在PFA并使用经典结果下，任何$ω_1$ -COMPACT本地紧凑的近似I型I型空间都包含一个非lindelöf子空间窄（实际上是$ω_1$的副本），而Suslin树则没有。其中有一个基本离散的子空间狭窄的空间。最后，我们研究（功能上）狭窄子空间的自然部分订单，以及这些订单为$ω$ - 或$ω_1$ clucted。
An open chain cover $\{U_α: α\inκ\}$ ($κ$ a cardinal) of a space $X$ is a systematic cover if the closure of $U_α$ is contained in $U_β$ when $α<β$, and $X$ is Type I if $κ=ω_1$ and the closure of each $U_α$ is Lindelöf. A closed subspace $D\subset X$ is narrow in $X$ if for each systematic cover $\{V_α: α\inω_1\}$ of $X$, either there is $α$ such that $D$ is included in $V_α$, or the closure of $V_α$ intersected with $D$ is Lindelöf for each $α$. Taking systematic covers given by preimages by $s$ of $[0,α)$ for a continuous $s: X\to \mathbb{L}_{\ge 0}$ (where $ \mathbb{L}_{\ge 0}$ is the longray) defines functionally Type I spaces and functionally narrow subspaces. For instance, $ \mathbb{L}_{\ge 0}$ and $ω_1$ are narrow in themselves and any other space. We investigate these properties and relative versions, as well as their relationship, and show in particular the following. There are functionally Hausdorff Type I spaces which are not functionally Type I while regular Type I spaces are functionally Type I. We exhibit examples of spaces which are narrow in some but not in other spaces. There are subspaces of a Tychonoff space $Y$ that are functionally narrow but not narrow in $Y$, while both notions agree if $Y$ is normal. Under PFA and using classical results, any $ω_1$-compact locally compact countably tight Type I space contains a non-Lindelöf subspace narrow in it (a copy of $ω_1$, actually), while a Suslin tree does not. There are spaces with subspaces narrow in them that are essentially discrete. Finally, we investigate natural partial orders on (functionally) narrow subspaces and when these orders are $ω$- or $ω_1$-closed.
