A spacetime singularity is a place in spacetime such that it’s possible to reach the it from the outside within *finite* proper time. That mean, from the perspective of the object moving toward the singularity, only finite time had passed, and the object just move into non-space, somehow.
Singularity are generally understood to be the failure of physical theory at that range, because it’s unbelievable that objects just go into non-space in finite time. Maybe there are some extra variables that need to be accounted for, they are too small at bigger range but really matter once you shrink down to a small region around a point; a famous case of this is the Ultraviolet catastrophe which lead to quantum mechanics (very small stuff need new physical law), and quantum mechanics run into this kind of problem often. However, in relativity, an alternative approach is to just keep the same theory, push it to its logical conclusion, but still fix the singularity, by extending the solutions: adding more spacetime in, more than originally conceived, to complete spacetime. This procedure is analogous to you seeing a hole on a piece a paper and gluing some paper over it. This process leads to prediction of extra universe.
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