First let me preface by saying this question will be very math heavy, and might be as equally suited to math stack exchange as world building. For that I apologize, but since the reason for my question is world building, I've placed it here.
In the Universe I'm trying to build, there is the Mundane world (where humans live and things more or less follow the set, natural laws of physics), and the Divine world (where Gods live, and things are slightly less rigid). Both worlds are "infinite," but the Mundane world is still "bounded" by the Divine world.
This is where the problem with my definition begins. If both worlds are infinite, how can one be "bound" by the other (as in contained or embedded within)?
In math, there is the concept of countable and uncountable infinite sets, and, quite non-intuitively, one infinity can be "greater" than another infinity, or can even contain that infinity entirely. I'm looking for a similar concept for my Mundane and Divine spaces, in other words, the Mundane, physical universe is infinite, and yet, still contained within the larger, "more infinite" Divine universe.
The first idea that I thought of, and which I'm sure someone will offer as an answer, is to simply embed our 3-dimensional physical universe (Mundane world) into a "higher dimensional" spiritual universe (Divine world). But this feels like side-stepping the real question and seems very much like a cheap way out. As such, I won't be giving points for such an answer. I'm not fully ruling out a 4 or 5 or whatever dimensional universe, but such a universe needs to allow a way for a 3-dimensional and infinite "Divine Space" to enclose a 3-dimensional and infinite "Mundane Space."
The Universe we know does all kinds of things that seem to "break common sense math," such as Quantum Renormalization and String Theory relying on the axiom that the sum of all natural numbers is -1/12, or Gabriel's Horn, which has infinite surface area in a finite volume. Yet all of these strange things are mathematically valid, and in some cases must be true given experimental evidence for how our physical Universe works, and I'm looking for a similar approach here.
So my question is:
Without simply embedding it in a higher dimensional space, is there a way, mathematically, to describe an infinite space bounded by another infinite space?
Edit: Here is a qualitative, though certainly not quantitative (or mathematically rigorous) explanation of a partial approach:
Imagine we somehow have an infinite hypersphere, one which contains an infinite 3d volume, and we look at it as a projection into 3d space. When you project a hypersphere into 3d space, you get something resembling two spheres inside each other. In this case, the "outer sphere" is the divine world, with its surface facing inwards towards the inner sphere, and the "inner sphere" is the mundane world, with its surface facing outwards towards the inner sphere. Both surfaces are actually 3-dimensional spaces, and both are infinite, yet in a higher spatial dimension, the outer surface "contains and bounds" the inner surface, with the extra dimensionality of the hypersphere being used to "fold" the infinite Mundane Universe so that it is contained inside the infinite Divine Universe.
As an aside, it is trivial to generalize and say a hypersphere contains an "infinite number" of 3d spheres, in the same way a plane contains an infinite number of lines or a cube an infinite number of planes, but this is the "just throw away a dimension" approach I'm trying to avoid, plus in this generalization the 3-d sphere is finite in extent.
What we need is, like the Gabriel's horn approach, to come up with a finite space containing an infinite smaller 3-dimensional space, and that finite space itself being able to be embedded an infinite space of the same dimensionality.
I'm not sure this makes sense, but then higher dimensional thinking never really does to brains evolved to watch out for lions in 3 spatial dimensions + 1 of time... ^^; Is there a mathematical definition closely resembling what I've described?
Edit2: As another aside, perhaps to give some background which may help in coming up with a more satisfying answer, the thought experiment which triggered my asking this question in the first place is this:
In my story, the Divine world and Mundane world cannot interact directly, and must do so through an intermediary "quasi-world" which allows one to pass between them.
This lead me to asking the question that, in this quasi-world (which resembles a forest but has a sky and sun), if you took a rocket and flew straight up, how far would you go and what would you see? Assuming you found you were on a planet, how far would space in that quasi-world go?
This further lead me to asking, if the Mundane World necessarily had to be contained within the Divine World, how could you ensure the Mundane World is infinite, yet still contained within the Divine? Presumably the point at which they join would be the quasi-world, and this would be finite, but the two worlds themselves would also be infinite, with the Mundane contained within the Divine.