Is my intuitive understanding of physical infinity correct? Quite possibly I got the section wrong again but the question is about cosmology. When they talk about an infinite Universe, they do not mean that in such a Universe there are two objects, the distance between which is ∞ parsec? This means that although any distance between any two objects is finite, there is always a finite distance between the other two objects, which is greater. That is, all distances are always finite, but the greatest distance does not exist. Is this intuitive understanding correct?
 A: 
... all distances are always finite, but the greatest distance does not exist. Is this intuitive understanding correct?

Yes, that is correct. An "infinite" universe is shorthand for a universe in which all distances are finite, but there is no upper limit to the distance between two objects. The size of the observable universe places an upper limit on the distances that we can observe, so we cannot tell whether the universe is infinite, or is just larger than the observable universe but still finite.
A: Your understanding of the concept is quite on point, but I would like to add a little correction: to have an infinite universe it isn't strictly necessary to always have two physical objects at a greater distance than the distance $A$ previously measured; it is simply requested that two objects could be placed at a greater distance than $A$.
What I mean is: picture an infinite, flat, sheath of paper, with a bunch of dots drawn on a portion of it, and then just blank paper in all direction, with no dots. Well you can now see that it's reasonable to consider this sheath infinite in the 2D space, even if there are no dots at a greater distance than let's say $B$, and this is because we always can draw dots at a greater distance than $B$, regardless of how large $B$ is.
As a side note: keep in mind that we do not see an infinite universe, and probably, if the current theories are correct, we will never see one. For more on this see Hubble Volume and Observable Universe.
A: We have no experimental fact that universe or space is infinite. And also note that infinity is a mathematical concept and has little relevance in physical world. So pretty much anything that exists in space is at a finite and measurable distance from any other object. No matter how great, the distance will be measurable.
