Compact vs. Polish space in application field. I was asked in a presentationto give an example of an engineering system where you want to use a system's state space as a Polish space but not a compact space. The audience was all electrical engineers. I was the only one having Mathematics background. I was presenting a generic Markov chain model, where I took the state space of the Markov chain as a Polish space but not a compact state space.
So they wanted to know why a compact space is not enough, why you need a Polish space.  Could anyone give me some supportive answers in this regard? A concrete example might work too. Thank you.
 A: This is not an engineering site, but I can give you a mathematical answer.
Here are two examples of topological spaces which occur in Markov chains, and which exhibit the transition between compact and Polish.

*

*the space $\{1,2\}^\mathbb N$, i.e. infinite sequences of $1$'s and $2$'s, is compact;

*the space $\mathbb N^\mathbb N$, i.e. infinite sequences of arbitrary natural numbers, is Polish and noncompact.

In both cases one is using the product topology with a discrete base space, either $\{1,2\}$ or $\mathbb N$. Note that $\{1,2\}$, like any finite space, is compact; whereas $\mathbb N$, like any unbounded space, is not compact.
Compactness of $\{1,2\}^\mathbb N$ is a consequence of Tychonoff's Theorem, a general theorem saying that any product of compact spaces is compact. Noncompactness of $\mathbb N^\mathbb N$ is more or less an elementary exercise in the definitions: roughly speaking, the fact that the first coordinate $n_1$ of a general element $(n_1,n_2,n_3,...) \in \mathbb N^\mathbb N$ can take on an unbounded set of values implies that the space $\mathbb N^\mathbb N$ itself is noncompact.
Regarding the formalism of Markov chains, all of the spaces involved in these examples may be regarded as metric spaces: $\{1,2\}$ and $\mathbb N$ are both discrete metric spaces; and the product topologies on $\{1,2\}^\mathbb N$ and on $\mathbb N^{\mathbb N}$ are both metrizable. So these examples fit perfectly well into a formal theory of Markov chains based on metric spaces.
I would even venture to say that $\{1,2\}^\mathbb N$, or perhaps $\{H,T\}^\mathbb N$, is one of the very first examples of a Markov chain: it is a mathematical model of flipping a coin infinitely many times. Similarly, regarding any Markov process consisting of infinitely any repetitions of an experiment that has finitely many outcomes, that Markov process is compact.
For any experiment which one might model by an infinite, discrete probability space, one can model infinite repetition of that experiment with the noncompact Markov chain $\mathbb N^\mathbb N$. An example might be counting the number of widgets which fall out of the back of the widget truck in the next minute. Although that example might be of more interest to professional thieves than to electrical engineers, I suspect that after hearing that example your electrical engineers might be able to conjure their own electrical engineering example.
One last mathematical point: constructing a complete metric on $\mathbb N^\mathbb N$ is rather fun, but what's even more fun is that $\mathbb N^\mathbb N$ turns out to be "the same as" the irrational numbers (replace those scare quotes by "homeomorphic to" to get a proper mathematical statement). So the irrational numbers, despite the evident fact that their ordinary metric $d(s,t)=|s-t|$ is not complete, nonetheless can be assigned a complete metric that generates their ordinary topology.
Regarding how you might explain this to engineers, I can't help you much there! But I'll make two points. First, those engineers weren't wrong: Polish spaces are so named because the underlying mathematical theory was first developed by mathematicians in Poland. Second, don't underestimate their interest in mathematics: I suspect many of them were instilled with a love of math from their early education.
