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## ExampleaswritteninHubbard

A 2-dimensional complex manifold that is not second countable

We will describe a connected complex manifold of dimension 2 that is not second countable. This manifold is a close analog of Example 1.3.1 [which was the classic non-second countable surface], but the elementary "cut and past" approach used there doesn't work so well in higher dimensions, so we will instead use a description in terms of blow-ups. (For blow-ups, see Hatshorne, Shafarevich; see page 30 of Thurston's Three-Dimensional Geometry and Topology for an informal introduction. But readers who don't know about blow-ups can skip this example; it has no further applications in this book).

In $\mathbb{C}^2$, we will blow up every point of $\mathbb{C}\times${0}. More specifically, for any finite subset $Z\subset \mathbb{C}$, we denote by $$\widetilde{\mathbb{C}_Z^2}$$

the blow-up of $\mathbb{C}^2$ at all the points of $Z$, and set $X:=\mathop{\lim}\limits_{\mathop{\leftarrow}\limits_Z} \widetilde{\mathbb{C}_Z^2}$, where the finite subsets are partially ordered by inclusion. (For inverse limits, see Hatcher). There is a natural map $p:\widetilde{\mathbb{C}_Z^2} \rightarrow \mathbb{C}^2$.

This space $X$ is not a manifold, the inverse image $Y:=p^{-1}(\mathbb{C}\times${0}) consists of the disjoint union $Y_1$ of uncountable many copies $\mathbb{P}^1_z$ of $\mathbb{P}^1$, one for every $z\in \mathbb{C}$, and some horrible set $Y_2=Y-Y_1$. The set $Y_2$ is not closed; it accumulates on exactly one point of each $\mathbb{P}_z^1$, namely, the point corresponding to the horizontal direction through $z$.

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