"On $\Delta$-spaces and spaces $C_p(X)$"Kąkol, JerzyA topological space $X$ is called a \emph{$\Delta$-space } (weakly $\Delta$-space) if for every decreasing sequence $(D_n)_{n}$ of (countable) subsets of $X$ with $\bigcap_{n}D_{n}=\emptyset$, there is a decreasing sequence $(V_n)_{n}$ of open subsets of $X$, $D_n \subset V_n$ for every $n \in N$ and $\bigcap_{n}V_{n}=\emptyset$. Research about $\Delta$-spaces is strictly connected with a study of $\mathbb{Q}$-sets, one of the most mysterious objects in $\mathbb{R}$. We proved that $X$ is a $\Delta$-space if and only the dual of $C_p(X)$ endowed with the topology of the uniform convergence on $C_p(X)$-bounded sets carries the finest locally convex topology. This analytic approach provided several new results about $\Delta$-sets and $\Delta$-spaces. Every $\check{C}$ech-complete $\Delta$-space is scattered and every scattered Eberlein compact space is a $\Delta$-space. Nevertheless, compact scattered spaces $X$ not being a $\Delta$-space do exist, for example $X=[0,\omega_{1}]$. Every metrizable scattered space is a $\Delta$-space. A compact space $X$ is a weakly $\Delta$-space if and only if $X$ is scattered (Kurka). |
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