The Friedmann model of cosmology

Alexander Friedmann's model of cosmology(1922) assumes the universe consists of a homogeneous and isotropic distribution of gravitating point sources governed by Einstein's general theory of relativity. These assumptions are also made in Einstein's cosmological model. 

Einstein added a cosmological constant to his equation of gravity to develop a static model of the universe. Without the cosmological constant, the equation shows the universe must expand. Friedmann, unlike Einstein, was not wedded to the idea that the universe should be static and explored the consequences of not using the cosmological constant.

Friedmann's model has three variations, which differ between the way they expand and how space curves.

• Closed. The universe expands slowly enough for mutual gravitational attraction to slow the expansion and cause a contraction. Space is closed and finite, like the earth's surface.
• Open. The universe expands so quickly that mutual gravitational attraction cannot slow the expansion enough to cause a contraction. Space is infinite and bends like a saddle.
• Flat. The universe expands as in the open variation, but if the expansion were any slower it would collapse like a closed universe. In this variant, space is flat and infinite, like a sheet of paper that has no boundry.

Although respecting the mathematics, Einstein did not accept the physical validity of Friedmann's solutions. Friedmann died in 1925, before he saw the main prediction of his model validated by experiment. The Friedmann model of cosmology predicted that the speed at which any two galaxies move apart is proportional to the distance between them. So it predicted that the red shift of a galaxy is directly proportional to its distance from us, exactly as Hubble observed.

In the closed variation, space is finite because gravity bends it. Like the earth's surface, it has no boundary. Also, time is finite - it has a beginning and an end. The end of the universe is predicted by Friedmann's model in all three variants.

Friedmann had briefly considered suggested the Universe might have emerged from a point-like singularity, this was the first Big Bang model.

Friedmann model, further reading:

A Brief History of Time by Stephen Hawking
The Road to Reality by Roger Penrose §27.11) calls the three variants of the Friedman model FLRW models, reflecting later contributors.
Big Bang by Simon Singh p.151-156.
A Short History of Nearly Everything by Bill Bryson