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0. Introduction The structure and solidarity properties of general Markov chains satisfying the measure-theoretic condition of 59-irreducibility, for some <p, are now well known (see, for example, Orey [8], Revuz [11]), and show a striking similarity to these properties for irreducible Markov chains on a countable state space. If the state space is topological then the condition of <pirreducibility is somewhat unnatural; however, it has so far proved difficult to achieve results of the same strength when 53-irreducibility is replaced by other, more topologically motivated, conditions. Several authors have given results bearing some resemblance to the '99-solidarity' properties for Markov chains whose transition probabilities are weakly continuous (see, for example, Foguel [2], Jamison [5], Rosenblatt [13, 14]). In this paper we propose a different form of continuity condition, namely the existence of a continuous component in the transition mechanism of the chain. Our condition is automatically satisfied by strongly Feller chains, i.e. in the special case where the transition probabilities are strongly continuous; but in general it is much weaker than the strong Feller condition. In § 4 we shall show that if this 'continuous component' is non-trivial in suitable ways then certain local topological irreducibility, recurrence or positivity conditions imply the corresponding ^-solidarity properties. These results are analogous to those known for 'spread-out' random walks, and we show in §5 that in fact our continuous component exists for random walks if and only if the increment distribution is spread-out; hence our condition can be seen as an exact generalization of this random walk concept for more general Markov chains. In § 6 we turn to chains which do not necessarily have any topological irreducibility structure. The main theorem of this section, and of the whole paper, states that, when the chain has a suitably non-trivial continuous component and the topology is Tx with a countable basis, the state space admits a decomposition into a countable number of stochastically closed Harris sets (i.e. the restriction of the chain to each

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