We say that a topological space $X$ is sequential if the following holds : If $U$ is sequentially open then $U$ is open. By sequentially open we mean that $x \in U$ and $x_n \to x$ implies that $x_n$ lies eventually in $U$.
My questions:
- Is there a characterization of sequential spaces ?
- Can one replace the phrase "nets" by "sequences" everywhere ? For example: A function is continuous iff $f(x_n) \to f(x)$ whenever $x_n \to x$ or a subset $K \subset X$ is compact iff each sequence in $K$ has a convergent subsequence.
- How does one prove that $B(H)$ is not sequential in the weak / strong topology, where $H$ is a separable Hilbert space and $B(H)$ the set of bounded operators acting on $H$.