molecular biology - Is the eukaryotic nucleus composed of a single or double membrane?

The nucleus is topologically single membrane but functionally, and as visualized, double membrane. Naturally, this is a bit confusing.

First consider the mitochondria and plastids. One of these organelles has two entirely separate lipid bilayers, one of which is nested inside the other. If you start inside the organelle and draw a line to an external point, the line will pass through two lipid bilayers (assume there are no membrane folds to be encountered), first through the nested one and then through the entirely separate outer bilayer.

The nucleus at a gross level appears to be structured similarly. A line from the interior to the exterior will usually pass through two lipid bilayers, one that is the inner wall of the nucleus, then through the outer wall. This may seem like the inner wall is separate from, and nested inside, the outer wall, but it is not. As nicely illustrated in your reference (Martin 2005), the two walls are continuous with each other and not nested. Because of this, a line from nuclear interior to exterior may in fact not pass through any lipid bilayers; it may pass though a nuclear pore. Topologically (mathematically) speaking, the two walls are one surface, and the "inside" of the nucleus is on the same side of the surface as the outside. (The topological interior is perinuclear space and the interior of the endoplasmic reticulum.) Functionally though, the structure acts as a double membrane since it physically constrains its contents.

The distinction is not very important for most biological contexts, but for a few purposes needs to be considered. The main reason to pay attention to the distinction is that it probably is related to the evolutionary history of the nucleus: it is likely to have evolved from a single bilayer somehow. In contrast, the mitochondria and plastids, with nested bilayers, evolved from two bilayers, one from a separate cell and one from an enveloping piece of a host cell. This is why the distinction is important in the cited paper, which concerns the evolution of the nucleus.