  Misner (1967) space is a portion of 2-dimensional Minkowski spacetime,
identified under a boost $\mathcal B$. It is well known that the maximal
analytic extension of Misner space that is Hausdorff consists of one half of
Minkowski spacetime, identified under $\mathcal B$; and Hawking and Ellis
(1973) have shown that the maximal analytic extension that is non-Hausdorff is
equal to the full Minkowski spacetime with the point $Q$ at the origin removed,
identifed under $\mathcal B$. In this paper I show that, in fact, there is an
infinite set of non-Hausdorff maximal analytic extensions, each with a
different causal structure. The extension constructed by Hawking and Ellis is
the simplest of these. Another extension is obtained by wrapping an infinite
number of copies of Minkowski spacetime around the removed $Q$ as a helicoid or
Riemann surface and then identifying events under the boost $\mathcal B$. The
other extensions are obtained by wrapping some number $n$ of successive copies
of Minkowski spacetime around the missing $Q$ as a helicoid, then identifying
the end of the $n$'th copy with the beginning of the initial copy, and then
identifying events under $\mathcal B$. I discuss the causal structure and
covering spaces of each of these extensions.
