Questions regarding a point in space-time where matter is infinitely dense.

In general relativity there are two types of singularities. Coordinate singularities are associated with a specific way of ascribing space-time coordinates to each event, and refer to events where one of the coordinates becomes infinite as your approach it. For instance the event horizon of a black hole is of this kind, and lies infinitely far in the future in the natural coordinates of a distant observer, but is a perfectly normal point in the coordinates natural to an infalling observer. Curvature singularities are points in the topological closure of space-time, such that the curvature of space-time (a scalar quantity independent of coordinates) tends to infinity as you approach them.