I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces of $\mathbb{C}^n$ with $\dim(y^i)=i$ for all $i=1,\ldots,n-1$, and it is clear that $\mathrm{PSL}(n,\mathbb{C})$ acts transitively on these.
However, when the article discusses continuity of the action, I'm not sure what to make of it. By the orbit-stabilizer theorem, I know the space of flags (denoted by $Y$) is in one-to-one correspondence with $\mathrm{GL}(n,\mathbb{C})/\mathrm{UT}(n)$ where $\mathrm{UT}(n)$ is the group of complex upper triangular matrices. The article then claims that the action of $\mathrm{PSL}(n,\mathbb{C})$ on $Y$ is continuous with respect to both the "transcendental topology" and the Zariski topology. The transcendental topology is probably the quotient topology, induced by the above identification, but even in doing so I am not sure at all how to prove that the action is indeed continuous with respect to these topologies.
Does anybody have a hint?