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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.

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    $\begingroup$ Multiple metrics induce the same topology, but multiple topologies are never induced by the same metric. To put it another way, (metrizable) topologies can be considered as equivalence classes of metrics, under an equivalence relation called topological equivalence. $\endgroup$ – Keshav Srinivasan May 12 at 19:29
  • $\begingroup$ my main source of confusion is that the plain and torus both admit a flat metric. So given a flat metric, how can I decide whether the topology is a $T^2$ or $\mathbb{R}^2$? $\endgroup$ – B. T. May 12 at 19:50
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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.

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