a Polish space
/ POH-lish /
Throughout this field the standing assumption on the state space S is that it is Polish. This is not pedantry: the cleanest forms of Prokhorov's theorem, the Skorokhod representation theorem, the existence of regular conditional distributions, and the disintegration of measures all require exactly the structure a Polish space provides. Naming and understanding this class of spaces is what lets the abstract theory apply uniformly to R^d, to C[0,1], to D[0,1], and to spaces of measures themselves.
A topological space is Polish if it is separable (has a countable dense subset) and completely metrizable (admits a metric, compatible with its topology, under which it is a complete metric space). Note the subtlety: completeness is a property of a metric, but Polishness asks only that SOME compatible complete metric exist, so the open interval (0,1) is Polish (homeomorphic to R) even though its usual metric is not complete. Standard Polish spaces include all of R^n, any separable Banach or Hilbert space, the continuous-path space C[0,1] with the sup metric, the cadlag space D[0,1] with Billingsley's Skorokhod metric, and, crucially, P(S) the space of probability measures on a Polish S, with the Levy-Prokhorov metric. The class is closed under countable products and under taking closed subsets and Borel subsets (the latter giving the standard Borel structure).
Two consequences explain why probabilists insist on it. First, Ulam's theorem: every finite Borel measure on a Polish space is tight (inner regular by compact sets), which is the hypothesis that makes Prokhorov's converse hold and that lets approximation-by-compacts arguments run. Second, on a Polish space all uncountable standard Borel spaces are Borel-isomorphic to R (the Borel isomorphism theorem), which is what guarantees regular conditional probabilities and disintegrations exist, the technical backbone of conditional expectation done rigorously and of building stochastic processes. The honest point: drop separability or completeness and these guarantees can fail (non-tight measures, no regular conditional distribution), so 'Polish' is the minimal hygiene assumption under which the whole theory is clean.
C[0,1] with the uniform metric is Polish: it is complete (uniform limits of continuous functions are continuous) and separable (polynomials with rational coefficients are dense by Stone-Weierstrass). This is exactly why Donsker's theorem can invoke Prokhorov's theorem and the Skorokhod representation on C[0,1] without any extra hypotheses.
C[0,1] is Polish, which is why the full weak-convergence machinery applies to path-valued limit theorems.
Polish means SOME compatible complete metric exists, not that a given one is complete. (0,1) is Polish despite its usual metric being incomplete; what matters is the topology, not a particular metric.