Definition:Weakly Open Set

Definition

Let $K$ be a topological field.

Let $X$ be a topological vector space with weak topology $w$.

Let $U \subseteq X$.

We say that $U$ is weakly open (or $w$-open) in $X$ if and only if $U$ is open in $\struct {X, w}$.

Linguistic Note

By analogy, an open set of $X$ might be called "strongly open"

Indeed this is common if $X$ is say, a Fréchet space.

However, in general locally convex spaces, "strong topology" has a more specific meaning, so its use may cause confusion.

It is therefore preferable to call the topology on $X$ the original topology, and open sets of $X$ originally open.