Hi guys! So this question comes from the definition of space-times in both Hawking and Ellis "Large Scale Structure of Space-time" and Malament "Topics in the Foundations of General Relativity and Newtonian Gravitation Theory". My question in particular revolves around the fact that both texts assume a priori that space-time manifolds are connected (Hawking and Ellis p.56 and Malament p.103). For background: here connected is being used in the topological sense i.e. a topological space ##X## is connected if the only clopen subsets of ##X## are ##X## and ##\varnothing ## (well one can also define a connected space as a space which cannot be written as the union of two disjoint non-empty open sets but one can easily show that this definition is equivalent to the previous one). For topological manifolds, connectedness can be shown to be equivalent to path connectedness (every topological manifold is locally path connected and for locally path connected spaces, connectedness is equivalent to path connectedness). By path connectedness, one means that for any two points ##p,q\in M## there exists a continuous map ##f:I\rightarrow M## such that ##f(0) = p, f(1) = q##. So we are taking space-times to be such that any two points can be connected by a continuous path (the 0th homology group vanishes). A general topological space can be partitioned into its connected components i.e. the maximal connected subsets of the space (connectedness of subsets is with respect to the subspace topology). I personally can't imagine simply using local physical experiments to justify a global feature like connectedness/path connectedness of space-time. Malament does not elucidate on why we take space-time to be connected and Hawking states "The manifold ##M## is taken to be connected since we would have no knowledge of any disconnected component." What does he mean by this? Why would we not have knowledge of the connected components of space-time? Is he alluding to what I said before regarding how local physical experiments would be inadequate in determining if space-time was connected or not (since this is a global property)? But even if we couldn't actually determine it, why would that allow us to take space-time to be connected over the possibility of it having non-trivial connected components?