In an ancient thought exercise called the Dichotomy Paradox, a finite distance is divided in half an infinite number of times. Space has long been thought to be infinitely divisible in this way.
Once it was thought that matter, too, is infinitely divisible, but the concept of atoms (and later, of the quarks that make up subatomic particles) denies that idea. Matter is now known to come in infinitesimally small, indivisible packages. And loop quantum gravity propose that spacetime is the same way. According to loop gravity, spacetime has a fine structure consisting of "atoms" of spacetime which cannot be subdivided and which can contain only a finite amount of matter-energy.
The Big Bang theory proposes that just before the "explosion" in which the universe began expanding, all matter was compressed into an infinitely small point. If space has a fine structure composed of indivisible, finite quanta of length and volume, this infinite compression would not be possible.