7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
We may assume that U and V are intersections of finite collections
of sets of the form (3).
It follows that there is a set of the form (3) which contains x0 but
not y0.
I.e.
x0 ∈ {y ∈ X : |l(y) − l(x)| < ǫ}
while |l(y0) − l(x)| ≥ ǫ for some x ∈ X,l ∈ M and some ǫ > 0.
Then l(x0) = l(y0), and we conclude that M separates the points
of X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems