Topology of the Misner Space and its g-boundary

The Misner space is a simplified $2$-dimensional model of the $4$-dimensional Taub-NUT space that reproduces some of its pathological behaviours. In this paper we provide an explicit base of the topology of the complete Misner space $\mathbb{R}^{1,1}/boost$. Besides we prove that some parts of this space, that behave like topological boundaries, are equivalent to the $g$-boundaries of the Misner space.