Spaces with sharp bases and with other special bases of countable order

We study spaces with sharp bases and bases of countable order. A characterization of spaces with external bases of countable order is established (Theorem 2.7). Some necessary and sufficient conditions for a space X × S , where S is the convergent sequence, to have a sharp base are given (Theorem 3.2). It follows that a pseudocompact space X is metrizable iff X × S has a sharp base (Corollary 3.3). It is proved that a sharp base of finite rank is a uniform base (Theorem 4.4). Some other new results are also obtained, and some open questions are formulated.