Abstract:We prove that if $X^n$ is a union of $n$ subspaces of pointwise countable type then the space $X$ is of pointwise countable type. If $X^\omega $ is a countable union of ultracomplete spaces, the space $X^\omega $ is ultracomplete. We give, under CH, an example of a \v Cech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].
Keywords: ultracompleteness, \v Cech-completeness, countable type, pointwise countable type
AMS Subject Classification: Primary 54H11, 54C10, 54D06; Secondary 54D25, 54C25