If the universe is infinite, it has always been infinite. At the Big Bang, it was infinitely dense. Since then it has just been getting less dense as space has expanded. Imagine a large flat rubber sheet with sand placed as closely together as possible on the sheet. The rubber sheet is a 2-D representation of spacetime and the sand would be matter. Now imagine stretching that rubber sheet so it has a larger and larger area. The sand would be less dense. If the sand particles had sufficiently strong force(s) of attraction between them (analogous to gravitational, electromagnetic, strong nuclear or weak nuclear forces), despite the expansion of space, they might clump up into atoms/planets/stars/galaxies. Each of the galaxies however could be getting further and further from each other (assuming the gravitational attraction between galaxies is too weak to overcome the expansion of space).
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