Birkhoff and Peressini proved that if ( X , T ) (X,\mathcal {T}) is a complete metrizable topological vector lattice, a sequence converges for the topology T \mathcal {T} iff the sequence relatively uniformly star converges. The above assumption of lattice structure is unnecessary. A necessary and sufficient condition for the conclusion is that the positive cone be closed, normal, and generating. If, moreover, the space ( X , T ) (X,\mathcal {T}) is locally convex, Namioka [11, Theorem 5.4] has shown that T \mathcal {T} coincides with the order bound topology T b {\mathcal {T}_b} and Gordon [4, Corollary, p. 423] (assuming lattice structure and local convexity) shows that metric convergence coincides with relative uniform star convergence. Omitting the assumptions of lattice structure and local convexity of ( X , T ) (X,\mathcal {T}) it is shown for the nonnecessarily local convex topology T ru {\mathcal {T}_{{\text {ru}}}} that T b ⊂ T ru = T {\mathcal {T}_{\text {b}}} \subset {\mathcal {T}_{{\text {ru}}}} = \mathcal {T} and T b = T ru = T {\mathcal {T}_{\text {b}}} = {\mathcal {T}_{{\text {ru}}}} = \mathcal {T} when ( X , T ) (X,\mathcal {T}) is locally convex.