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Will the universe eventually collapse into a singularity?
The universe will never fall into a singularity, but it can be said that the universe's expanding phases are drawing closer to an anti-singularity.
In a certain sense, the universe is already in an anti-singularity. The observable universe has been expanding for some 13 billion years and will expand forever.
The most important implication of this is that the universe is expanding at an ever-increasing rate. The observable universe will expand until it becomes a perfect sphere, with space continuing to stretch forever.
As the universe expands, it cools. Eventually, all stars will cease to exist because there is not enough energy in the universe to sustain them.
Eventually, all that will remain in the universe is dark energy and (very) cold matter. The energy in the entire universe will be evenly distributed.
At this point, there will be no more expansion of space. The universe is already in a state of anti-singularity and the only thing left to do is for everything in the universe to eventually reach equilibrium with each other.