Summary: If \( S \) is a semigroup in \( \mathbb{R}^n \) that is not separated by a linear functional, then it is known that the closure of \( S \) is a group. We investigate a similar statement in an infinite dimensional topological vector space \( X \). We show that if \( X \) is an infinite dimensional Banach space, then there exists a semigroup \( S\subset X \), not separated by the continuous functionals supported by the closed linear span of \( S \), for which the closure of the semigroup is not a group. If \( X \) is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if \( X \) is \( \mathbb{R}^{\omega} \), the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as \( \mathbb{R}^{\infty} \), the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.