论文标题
Ascoli和依次的Ascoli空间
Ascoli and sequentially Ascoli spaces
论文作者
论文摘要
A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is evenly continuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the compact-open topology.研究了（依次）的Ascoli空间的各种特性，我们给出了依次的Ascoli空间的几个特征。加强了Arhangel'skii的结果，我们表明遗传性的Ascoli空间是Fréchet-urysohn。与Bohr拓扑的本地紧凑型Abelian Group $ G $是依次是Ascoli Iff $ G $紧凑。如果$ x $完全是紧凑的或接近依次紧凑的，那么它是一个顺序的ascoli空间。局部紧凑的空间和Ascoli空间的乘积是Ascoli。如果另外$ x $是$μ$ - 空间，则$ x $是本地紧凑的，如果$ x $的产品带有任何Ascoli Space是Ascoli空间。扩展[18]和[16]的主要结果之一，我们表明$ c_p（x）$是依次是ascoli iff $ x $具有属性$（κ）$。我们在$ x $上给出了必要的条件，该$ x $为$ c_k（x）$依次是ascoli。对于每一个可Metrizable Abelian Group $ y $，$ y $ -tychonoff space $ x $和非零可计数的序列$α$，bace $b_α（x，y）$ baire-$α$函数从$ x $ x $ to $ x $ to $ y $ is $ y $ y是$κ$ -FRUCHET--FRUCHET-redchet---freth-y-fréchet-sossohnand yres assascoli。
A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is evenly continuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet--Urysohn. A locally compact abelian group $G$ with the Bohr topology is sequentially Ascoli iff $G$ is compact. If $X$ is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally $X$ is a $μ$-space, then $X$ is locally compact iff the product of $X$ with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that $C_p(X)$ is sequentially Ascoli iff $X$ has the property $(κ)$. We give a necessary condition on $X$ for which the space $C_k(X)$ is sequentially Ascoli. For every metrizable abelian group $Y$, $Y$-Tychonoff space $X$, and nonzero countable ordinal $α$, the space $B_α(X,Y)$ of Baire-$α$ functions from $X$ to $Y$ is $κ$-Fréchet--Urysohn and hence Ascoli.
