关于有尖基空间的度量化
On the metrizability of spaces with a sharp base
摘要：
空间$X$的基$\mathcal{B}$被称为是尖锐的，如果只要$x\in X$和$(B_n)_{n\inω}$是$\mathcal{B}$的成对不同元素的序列，每个元素都包含$x$，则集合$\{\bigcap_{j\le n}B_j:n\inω\}$是$x$的局部基。我们回答了由Ajanhe等人和Arhangel $'$ ski \u Schmidt等人提出的问题，证明了具有尖基的伪紧吉洪诺夫空间不一定是可度量化的，并且具有尖基的空间与$[0,1]$的乘积不一定有尖基。我们证明了各种度量化定理，并提供了一个表征沿线的Ponomarev的点可数基地。
英文摘要：
A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n)_{n\inω}$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le n}B_j:n\inω\}$ is a local base at $x$. We answer questions raised by Alleche et al. and Arhangel$'$ski\uı et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and $[0,1]$ need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarev's for point countable bases.
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