Does the metric uniquely specifies the topology?

Question

As far as I know, given a metric, we can always find the induced topology. However, I was wondering if this topology is unique. To put it another way, metric completely encodes the local data about the manifold but does it also completely encode the global data about the manifold? My main motivation for this question is to understand whether the Einstein field equation uniquely determines the topology of spacetime. I apologize if my question is basic. Thank you so much in advance.

Answer 1

Every metric determines a well-defined topology. Also, some metrics are flat; others aren't. But flatness is not a global property, it is only a local property. Hence there is no reason to expect that objects with different global properties cannot have metrics with similar local properties. For example, the statement that both the plane and the torus have flat metrics is not a contradiction to the local nature of flatness.

To be explicit, a metric space is flat if it is locally isometric to Euclidean space. Another explicit way to say almost the same thing is that a Riemannian metric is flat if and only if the metric has the property that its coefficients satisfy a certain PDE; that property is certainly local.