[0] http://www.geom.uiuc.edu/~banchoff/Flatland/
Also, not having a border is not the same as being infinite. Zogg from Betelgeuse started a great series on the topic [1], but disappeared after two episodes. I highly recommend it! But, long story short, it's entirely possible to have a finite universe that, nonetheless, has no edge. Go far enough in one direction and you might end up back where you started.
[1] https://www.youtube.com/watch?v=_k3_B9Eq7eM
Assuming there's a physical edge to our universe, there's still the problem of the "observable universe" -- there's a lot more of the universe that we will simply never be able to see or reach, because it's moving away from us too quickly. That defines another kind of edge, one that's much "closer". Entities beyond our observable universe can't affect us gravitationally; it takes time for the effects of gravity to propagate from one point to another, as "gravitational waves", and the universe is inflating too quickly for those to reach us.
You might say that things further away from us, but within the observable universe, might be affected by things beyond, because they're "closer" to it than we are. I think that's correct, but by the same token, I think the information about those effects would never reach us in finite time.
That's why I was thinking in terms of extra spatial dimensions. You might have a bunch of matter "nearby" but totally inaccessible to you, because we're incapable of moving along the extra axis. But that gets back to the research I linked.
There's no reason to believe our "common sense" or "intuitive understanding" has any relevance to how the universe is or isn't.
According to an ant's common sense humans for sure have antennas somewhere on their bodies, even if not visible to an ant.
Now imagine there are two such spheres near to each other. Two beings whose physical laws and interactions were entirely confined to their respective surfaces would be unaware of each other and unable to influence each other in any way - gravity for example is only able to travel around the surface of the sphere, not into whatever 'space' lies in between. The surface of each sphere is its own universe.