= Solution
A <spacetime> is a smooth four-dimensional <manifold> equipped with a smooth <Lorentzian metric>, conventionally of <metric signature> $(-,+,+,+)$, and a choice of <time orientation>. The usual manifold assumptions include the <Hausdorff space> and <second-countable space> conditions. A physical model also specifies matter fields and requires the <Einstein field equations> and matter equations.
<Diffeomorphism invariance of general relativity> means that relabelling events by a smooth invertible map, while transforming the <metric tensor> and every matter field together, preserves the form of the equations. Passively, a coordinate change gives new components for the same geometric fields. Actively, pulling all fields back by a <diffeomorphism> gives another representative of the same physical geometry, subject to any prescribed boundary conditions or boundary symmetries.
\b[The geometry and matter together carry the physical content; coordinate labels do not.] A <diffeomorphism> need not be an <isometry> of a fixed metric, and unrelated metrics are not automatically physically equivalent.
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