论文标题
某些功能空间的拓扑特性
Topological properties of some function spaces
论文作者
论文摘要
令$ y $是一个至少包含两个点的METRIZABLE空间，让$ x $为$ y _ {\ Mathcal {i}} $ - tychonoff空间，用于某些理想的$ \ Mathcal {i} $ compact $ x $。用$ c _ {\ mathcal {i}}（x，y）$连续函数的空间从$ x $到$ y $ endosed $ \ mathcal {i} $ - 开放式拓扑。我们证明$ c _ {\ Mathcal {i}}（x，y）$是fréchet -urysohn iff $ x $具有属性$γ_ {\ Mathcal {i}} $。我们表征零 - 尺寸tychonoff spaces $ x $，为此空间$ c _ {\ mathcal {i}}}（x，x，{\ bf 2}）$是顺序的。扩展了Gerlits，Nagy和Pytkeev的经典定理，我们表明，如果$ y $不是紧凑的，则$ c_ {p}（x，y）$是fréchet -urysohn -urysohn iff，iff sequeention iff if sequeentif iff us $ k $ - space iff $ x $ x $具有属性$ $γ$。对于在局部凸出空间中采用值的有限连续函数的空间，获得了类似的结果。用$ b_ {1}（x，y）$和$ b（x，y）$表示Baire One功能的空间和所有Baire功能的空间分别从$ x $到$ y $。如果$ h $是$ b（x，y）的子空间，含有$ b_ {1}（x，y）$，则$ h $是可Metrizable如果是$σ$ - $σ$ - space如果有可计数$ cs^*$ - cartarable $ x $ if $ x $是可计数的。如果另外$ y $不是紧凑的，那么$ h $是fréchet -urysohn iff ifffréchet，如果是$ k $ - face -face -fack -iff iff，则具有可计数的紧密度，如果$ x _ {\ aleph_0} $具有属性$γ$，则$ x _ {\ aleph_0} $是$ x $ a $ x $ topicy。我们表明，如果$ x $是波兰空间，那么如果$ x $是可计数的，那么空间$ b_ {1}（x，\ mathbb {r}）$是正常的。
Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\mathcal{I}$-open topology. We prove that $C_{\mathcal{I}}(X,Y)$ is Fréchet - Urysohn iff $X$ has the property $γ_{\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\mathcal{I}}(X,{\bf 2})$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ is not compact, then $C_{p}(X,Y)$ is Fréchet - Urysohn iff it is sequential iff it is a $k$-space iff $X$ has the property $γ$. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by $B_{1}(X,Y)$ and $B(X,Y)$ the space of Baire one functions and the space of all Baire functions from $X$ to $Y$, respectively. If $H$ is a subspace of $B(X,Y)$ containing $B_{1}(X,Y)$, then $H$ is metrizable iff it is a $σ$ - space iff it has countable $cs^*$ - character iff $X$ is countable. If additionally $Y$ is not compact, then $H$ is Fréchet - Urysohn iff it is sequential iff it is a $k$ - space iff it has countable tightness iff $X_{\aleph_0}$ has the property $γ$, where $X_{\aleph_0}$ is the space $X$ with the Baire topology. We show that if $X$ is a Polish space, then the space $B_{1}(X,\mathbb{R})$ is normal iff $X$ is countable.
