A space is a k-space if for each , is closed in if and only if is closed in for all compact . A space is a sequential space if for each , is closed in if and only if is a sequentially closed set in . A set is sequentially closed in the space if whenever we have and the sequence converges to , we have . A set is sequentially open in if is sequentially closed in . In both of these definitions, we can replace “closed” with “open” and the “only if” part of the definition always hold. Thus in working with these definitions, we only need to be concerned with the “if” part. Every sequential space is a k-space. The converse does not hold. In this short note, we show that the converse holds if every point in the space is a -set. This is a basic fact about k-spaces. For other basic facts on k-spaces and sequential spaces, see the following:
In a given space , is a -set in if where each is open in , i.e. is the intersection of countably many open sets. A point is a -set in if the singleton is the intersection of countably many open subsets of . It is a well known fact in general topology that in a compact Hausdorff space , if is a -set in , then there is a countable local base at . It follows that if every point of a compact Hausdorff space is a -set in , then is first countable (see The cardinality of compact first countable spaces, II).
Let be a space in which every point is a -set in . Then if is a k-space then is a sequential space.
Proof. Suppose is not closed in . We show that is not sequentially closed in , i.e. there is a sequence such that and .
Since is a k-space and is not closed, there is a compact such that is not closed in . Every point of is a -set in and thus a -set in . It follows that is first countable.
Let such that (the closure is taken in ). Since is first countable, there is a sequence such that . This means is not sequentially closed in .