Closed topological embedding of complex algebraic varieties into a smooth manifold In the book "Representation Theory and Complex Geometry” by Ginzburg-Chriss page 93 they claim that (the analytification of) every complex algebraic variety admits a closed embedding in a smooth manifold. They do not really provide a reference for this statement. Why is this true?
For quasi-projective varieties, the claim is fairly obvious. But it seems to me that they do not impose this restriction. Moreover, I would say that we need the variety to be separated for this but (I can live with that).
 A: It took me ages to find this reference (and I actually found it in this post). There is a theorem by Acquistapace–Broglia–Tognoli in An embedding theorem for real analytic spaces stating:
Theorem. Let $X$ be a paracompact connected $n$-dimensional analytic space and suppose that $q := \sup_{x \in X} \dim T_x X < \infty$. Then $X$ admits a closed $C^\omega$-immersion into $\mathbb{R}^{n+q}$. 
This is a generalization of Grauert's embedding theorem for real analytic manifolds. From this we obtain:
Corollary. Let $X$ be a separated complex algebraic variety. Then $X^{\mathrm{an}}$ admits a topological closed embedding into $\mathbb{R}^N$ for some $N$.
Proof. $X^{\mathrm{an}}$ is a paracompact complex (thus also real) analytic space with finitely many connected components and with bounded tangent space dimensions. Now, apply the theorem.
Remark 1. The only thing that surprises me is why the ABT paper has only one citation on MathSciNet since 1979; I don't know if there's another source for the above theorem (I hope it's right actually).
Remark 2. I don't know if there isn't a simpler argument.
