Suppose that a finite-state Markov chain with transition matrix $M$ has *multiple* stationary distributions (or invariant measures). Let $\Pi$ denote the set of the stationary distributions.

Fix any $\pi \in \Pi$. Does there exist a Markov chain with transition matrix $M'$ with the following properties?

1. $M'$ is "close" to $M$.
2. $M'$ has a *unique* stationary distribution $\pi'$.
3. $\pi'$ is "close" to $\pi$.

Here, "close" means that 
each element of difference $\vert M' - M \vert$ is sufficiently close to zero; each element of difference $\vert \pi' - \pi \vert$ is sufficiently close to zero.

In other words, is it possible to select arbitrary one stationary distribution $\pi \in \Pi$ by slightly perturbing the Markov chain $M$?

How about countable- or uncountable-state Markov chains?