  We discuss how developments in physics often imply in the need that spacetime
acquires an increasingly richer and complex structure. General Relativity was
the first theory to show us the way to connect space and time with the physical
world. Since then, scrutinizing the ways spacetime might exist is, in a way,
the very essence of physics. Physics has thus given substance to the pioneering
work of scores of brilliant mathematicians who speculated on the geometry and
topology of spaces.
