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?