For what follows, let us restrict ourselves to discussing discrete probability distributions.
When learning about things like relative entropy (aka Kullback-Leibler divergence) $D(\cdot || \cdot)$, one typically learns that it does not technically define a metric, since $D(P||Q) \ne D(Q||P)$, for some distributions $P,Q$.
In general, I don't think there is any well-defined notion of distance between two probability distributions.
However, when learning about Markov chains, we learn about nice conditions under which some distribution $\pi_0$ converges to the stationary distribution of the Markov chain $\pi$ (assuming it exists).
Since there is no well-defined metric on the set of probability distributions, what is going on here? How is this convergence defined? In precisely what sense is $\pi_0$ approaching $\pi$? Is there some (obvious) underlying topology that I am just not seeing? Can someone help reconcile what's going on here?