Definition:Weakly Open Set/Linguistic Note

Linguistic Note on Weakly Open Set

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.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $3.11$: The weak topology of a topological vector space