Well a (topological) manifold is a topological space X, that is locally homeomorphic to some d-dimensional euclidean space R^d. A topology on a set X is a way o

Well a (topological) manifold is a topological space X, that is locally homeomorphic to some d-dimensional euclidean space R^d. A topology on a set X is a way of defining what points are "close together" in some sense, more precisely it is a collection of subsets of X, called the open sets with respect to the topology. The collection has to contain both the set X and the empty set, any union of sets in the collection and any finite intersection. Continuous maps between topological spaces are then maps f: X -> Y, that have the property that every f^{-1}(U) preimage of an open set U of Y is an open set of X. And homeomorphisms are continuous maps with continuous inverse. Euclidean spaces have a natural topology induced by the standard euclidean norm, whenever you have a notion of distance or metric d: X * X -> R defined on a space X you can speak of the "open balls" with respect to this metric, that is for a fixed point P all points P', such that the distance d(P,P') < c is smaller than some constant c. Those generate a topology. For some arbitrary topological space X to be locally homeomorphic to R^d, then just means that every point p of X has a open neighborhood, that maps homeomorphically to some open ball in R^d. That is to say it locally "looks like" euclidean space. One usually wants to do differential geometry on manifolds, that is have some notion of first/second/higher order geometry, in other words do all the things one knows and loves from calculus/analysis on manifolds. For that to work one need that coordinate changes are differentiable, that is given two open neighborhoods U and V of a point p in X and and homeomorphisms f: U -> U' in R^d and g : V -> V' in R^d one needs the composition g . f^{-1} to be differentiable. A stochastic manifold is then simply such a (differentiable) manifold together with a density p, that integrates to one $\int_X p = 1$.