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You can imagine the balloon’s curved surface geometry without assuming that it is embedded into a 3D space. This is what Riemann discovered in the 19th century. Such a non-embedded space is called a manifold. In the case of the balloon, it’s simply a non-Euclidian two-dimensional space. A third dimension doesn’t enter into the picture in that formulation.

Four-dimensional spacetime is considered such a manifold. Its three-dimensional space “slices” are predicted to expand with time by general relativity (first predicted by Alexander Friedmann in 1922), and actual observations have confirmed those predictions.