  A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in
X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of
$\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le
n}B_j:n\in\omega\}$ is a local base at $x$. We answer questions raised by
Alleche et al. and Arhangel$'$ski\u{\i} 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.
