论文标题
在空间$ c_0 $ in Spaces $ c_p（x \ times y）$的副本上
On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$
论文作者
论文摘要
Cembranos和Freniche证明，每两个无限的紧凑型Hausdorff Spaces $ x $和$ y $ y $ y $ the Banach Space $ c（x \ times y）$连续实现的函数上的$ x \ times y $ endowed endowed the supremum norm and the supremum norm and the supremum norm and coplement nord con con $ c _ c _ {0} $都包含了补充的副本。 We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of $\mathbb{R}^ω$ or a complemented copy of the space $（c_ {0}）_ {p} = \ {（x_n）_ {n \inΩ} \ in \ mathbb {r}^ω\ colon x_n \ to 0 \} $，都与产品拓扑结合在一起。我们表明，当$ x \ times y $是假发时，后一种情况始终存在。另一方面，假设连续假设（甚至是较弱的设定理论假设），我们提供了一个pseudoCompact Space $ x $的示例，以便$ c_ {p}（x \ times x）$不包含$（c_ {0}）$（c_ {0}）_ {p} $的补充副本。 作为第一个结果的推论，我们表明，对于所有无限的tychonoff，$ x $和$ y $ $ x $和y $ space $ c_ {p}（x \ times y）$都是与空间$ c_ {p}（p}（x \ times y）\ times \ times \ mathbb {r} $的同型同构的同型。 $ x $使得$ c_ {p}（x）$无法通过连续的线性陈述映射到$ c_ {p}（x）\ times \ times \ times \ mathbb {r} $。这为Arkhangel'ski问题的问题提供了积极的答案，该问题对于$ c_p（x \ times y）$的空间。 另一种推论指出，对于每个无限的Tychonoff空间$ x $和$ y $ the $ c_ {k}（x \ times y）$连续功能上的$ x \ times y $ en en赋予了紧凑型拓扑赋予的商品，该拓扑赋予了以下三个space of space of以下三个台面： $（c_ {0}）_ {p} $或$ c_ {0} $。
Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ endowed with the supremum norm contains a complemented copy of the Banach space $c_{0}$. We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of $\mathbb{R}^ω$ or a complemented copy of the space $(c_{0})_{p}=\{(x_n)_{n\inω}\in \mathbb{R}^ω\colon x_n\to 0\}$, both endowed with the product topology. We show that the latter case holds always when $X\times Y$ is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space $X$ such that $C_{p}(X\times X)$ does not contain a complemented copy of $(c_{0})_{p}$. As a corollary to the first result, we show that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ is linearly homeomorphic to the space $C_{p}(X\times Y)\times\mathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$ cannot be mapped onto $C_{p}(X)\times\mathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X\times Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_{k}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: $\mathbb{R}^ω$, $(c_{0})_{p}$ or $c_{0}$.
