Definition:Weakly Closed Set

Definition

Let $K$ be a topological field.

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

Let $C \subseteq X$.

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

That is, if and only if $U = X \setminus C$ is weakly open.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{V}$ Weak Topologies: $\S 1.$ Duality