  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.
