TITLE: When is Bialynicki-Birula decomposition a paving?
QUESTION [10 upvotes]: Suppose we have a smooth algebraic variety $X$ with an action of $\mathbb{C}^*$ with finitely many fixed points. Suppose $X$ can be covered by invariant quasi-affine open sets and suppose for each $x\in X$ the limit $\lambda x$ when $\lambda\to 0$ exists. Then Bialynicki-Birula proves that $X$ is a union of locally closed sets $W_i$ isomorphic to affine spaces.
I find often in the literature it is concluded that $X$ has a paving by affine spaces, but "paving" requires more: that $W_i$ are ordered in such a way that $\cup_{j \leq i} W_i$ is closed. But this issue is often ignored. Is there something I'm missing, for instance some general statement that would guarantee existence of such an ordering?

REPLY [3 votes]: Consider the partial ordering defined by
$$
i \le j
\qquad\text{if $\dim(W_i) \le \dim(W_j)$}.
$$
Since $\overline{W_i} \setminus W_i$ has dimension less than $W_i$ and is $\mathbb{C}^\times$-invariant, it is the union of some $W_j$ with $j < i$. Therefore,
$$
\bigcup_{j \le i} W_j = 
\bigcup_{j \le i} \overline{W_j}
$$
is closed.