A path in a topological space X is defined as a continuous function f from the closed interval [0,1] into X. A connected space X is defined as a space that cannot be represented as the union of two or more disjoint non-empty open subsets.
Neither of these definitions can be satisfied for discrete spaces. But what I'm wondering is whether there's a natural (and non-disjunctive) generalization of either of these notions that applies to both discrete and non-discrete spaces, and that generates results such as the following: in ℤ, the sequence (1, 2, 3, 4) is a "path" but the sequence (1, 3, 4, 2) is not, and the subspace {1, 2, 3, 4} is "connected" but the subspace {1, 2, 5, 6} is not.
I can see how one might construct analogues of the notions of path and connectedness that apply only to discrete spaces. But what I'm having trouble with is finding a generalization that subsumes the original notions, while also being satisfiable for discrete spaces in the manner above. I should mention also that I'd be happy to find a generalization that applies only to metric spaces.
Thanks for your help, and apologies for any unclarity in the question.
Any form of connectedness for topological spaces must by definition be a topological invariant, i.e. must be invariant under homeomorphism. More generally, if non-homeomorphic spaces can be distinguished by some property $\mathcal P$, then $\mathcal P$ must refer to other structural components than the topologies of spaces.
In your question you implicitly refer to the natural ordering on $\mathbb Z$. This has nothing to do with topology.
Thus, there is no concept of connectedness for discrete spaces. In fact, discrete spaces are uniquley determined by their underlying sets and there is no set-theoretic property other than the cardinality allowing to distinguish between sets. Therefore you must consider sets endowed with an additional structure not related to topology.
By the way, in your examples you do not consider sets , but sequences in ordered sets and it seems that you call such a sequence a path if its elements are arranged in ascending order. This is okay as a definition, but it has nothing to do with topology.