Closure of the graphic of the vector of the players' payoff functions

Question

Let $X_i$ be a compact subset of a topological vector space (not necessarily Hausdorff) , $i=1,\ldots,N$, $ \ u_i: X\rightarrow \mathbb{R}$ the payoff functions and $u$ the vector of the players' payoff functions.

Is true that the closure of the graphic of $u$ is a subset of $X\times \mathbb{R}^N$?