How does modern cosmology preclude the possibility of either a center or an edge to the universe?

The Cosmological Principle is one of the fundamental assumptions in modern cosmology, underpinning the understanding that the universe is a level playing field in terms of its structure and composition on the grandest scales. Visually, imagine looking through a kaleidoscope; no matter how you turn it, the patterns always look consistent and uniform. This is akin to the concept that on a sufficiently large scale, no matter where you are in the universe, you'd see a similar distribution of galaxies, stars, and cosmic material.
This principle means starkly that the universe favors no particular location or direction. When we ponder the implications, it's profound: if the universe looks the same from any standpoint and in all directions, we cannot speak of a central point from which everything else unfolds. This thought challenges the very human inclination to place ourselves at the center of our observations. By accepting a universe with no special vantage point, we let go of a 'center' as much as we dismiss the notion of an 'edge'—both ideas are relics of a less-informed age and have no basis in the cosmic reality unveiled by modern science.
The Cosmological Principle also supports the notion of an observable universe that is effectively the same for every observer, which has strong implications for how we study and interpret cosmological data. By shedding the misconception of a universe that revolves around a central point, scientists are better equipped to unravel the fundamental forces and the history of our cosmic habitat.

Edwin Hubble's landmark discovery, now known as Hubble's Law, is the linchpin that substantiates the idea of a dynamic, breathing cosmos. The law itself is beautifully simple in statement: all objects in the universe move away from each other with a velocity proportional to their distance. A useful metaphor is the baking of a raisin bread; as the bread expands in the oven, the raisins within move away from each other—not because any single raisin is the focal point, but due to the expansion of the dough itself.
The observation that galaxies seem to be retreating from our vantage point might hint at Earth as central, giving rise to the understandable yet illusory concept that we are some sort of cosmic nexus. But just as any raisin in the bread would see the same expansion, every galaxy observes similar recessions, dispelling the notion of a privileged central position. This principle is instrumental in conveying that our universe expands uniformly and does so without granting any spot the title of 'center'.
In essence, Hubble's Law provided not only observational evidence for a universe in flux but also kindled a revolution in thought, underscoring that we are not the bullseye in an astronomically archaic dartboard. Instead, we inhabit just one of the myriad galaxies adrift in this ever-expanding cosmic sea.

Grasping the concept of an expanding universe is akin to accepting that there's no end of the ocean when you're in the middle of it—with every mile you sail, there's only more sea. The expansion of the universe is central to modern cosmological theories, neatly tying into the observations encapsulated by Hubble's Law and the guiding philosophy of the Cosmological Principle.
Rather than expanding from a specific point or blowing up like a balloon with edges, the universe stretches in a manner that defies the traditional concept of an edge. It balloons outward in all directions, everywhere at once. This expansion is not into a pre-existing external space but an increase in the very fabric of space itself. Any talk of an 'edge' fails when we consider that the universe is not expanding into anything that otherwise borders it.
This 'Edgeness' notion collapses under the weight of Einstein's general theory of relativity, which postulates a universe with a curved, complex topology. It is tempting to visualize traveling in a straight line to eventually hit a boundary, but in our actual universe, such a journey would potentially loop back to the origin in a phenomenon mirroring circumnavigation on Earth. The lack of an 'edge' is a testament to the elasticity of our cosmological theories, which allow us to view the universe not as an object within space but as a self-contained, unfurling phenomenon without the conventional confines of an interior or exterior.