# Union of Bruhat cells is open in $G$

Let $$G$$ be (the points of) a connected, reductive group over a local field $$k$$ with maximal split torus $$S$$, minimal parabolic $$P_0$$, and Weyl group $$W = N_G(S)/Z_G(S)$$. Let $$\Delta$$ be the corresponding set of simple roots, and for $$\Theta, \Omega \subset \Delta$$, let $$P_{\Theta}, P_{\Omega}$$ be the corresponding parabolic subgroups. Let $$C(w) = P_{\Theta}wP_{\Omega}$$, so that $$G$$ is the disjoint union

$$\bigcup\limits_{w \in W_{\Theta} \backslash W/W_{\Omega}} C(w)$$

Define a partial ordering on $$W_{\Theta} \backslash W/W_{\Omega}$$ by setting $$w < w'$$ if $$C(w) \subseteq \overline{C(w')}$$. Let

$$G_w = \bigcup\limits_{w < w'} C(w')$$

Is this an open set in $$G$$? This is claimed in Casselman's notes on representation theory, but I don't see why this should be the case.

Okay now that I have thought about this some more, it actually falls out pretty formally. The cells $$C(w_0)$$ are the orbits of the group $$P_{\Theta} \times P_{\Omega}$$ acting on $$G$$, so each closure $$\overline{C(w')}$$ is a disjoint union of cells $$C(w_0)$$. By definition of the order relation, $$\overline{C(w')}$$ is the disjoint union of the $$C(w_0)$$ with $$w_0 < w'$$. Now
$$G - G_w = \bigcup\limits_{\substack{w' \in W_{\Theta} \backslash W/W_{\Omega}\\w \not< w'}} C(w')$$
$$\overline{G - G_w} =\bigcup\limits_{\substack{w' \in W_{\Theta} \backslash W/W_{\Omega}\\w \not< w'}} \overline{C(w')} = \bigcup\limits_{w \not< w'} \bigcup\limits_{w_0 < w'} C(w_0)$$
If we take one of the cells $$C(w_0)$$, for some $$w' \not< w$$ and $$w_0 < w'$$, then of course $$w$$ is not less than $$w_0$$, so $$C(w_0)$$ is contained in $$G - G_w$$. Thus $$G - G_w$$ is equal to its closure.