Assumption of our universe being the surface of a 4-D sphere in order to describe relative uniformity in the cosmic microwave background?

It seems as if this is necessary to assume in order for the CMB to make an sense. Otherwise, an instantaneous infinite Euclidean space model would work. The issue is, with any visualization of the Big Bang, let's take a Big Bang visualization in 3 - dimensions, the radiation emitted from the creation of hydrogen atoms would not necessarily be all around us, as it is. Our position in space would dictate what we would observe in terms of background radiation.

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Do you have some linkable sources for that assumption or is it a thought of your own. IMHO your last sentence is wrong in the currently widely accepted models: Our position in space does not matter, because when the big bang happened all of the space we have today was tightly clumped together and then expanded so in a sense we are still inside of the big bang that seems to happen "everywhere" and because of that that radiation is all around us. – DrCopyPaste Apr 17 '14 at 9:26

To a high degree of accuracy, the cosmic background radiation we observe is homogeneous and isotropic... except for the CMBR dipole anisotropy, which is there because we are moving at $369\pm 0.9\;\mathrm{km/s}$ relative to the CMB rest frame, in which this anisotropy would vanish.

Relevantly, the Friedmann-Robertson-Walker family of solutions of general relativity used to model the Big Bang cosmology assume that the universe is, on the large scale, homogeneous and isotropic.

It seems as if this is necessary to assume in order for the CMB to make an sense. ... Our position in space would dictate what we would observe in terms of background radiation.

That's not the case. There are four distinct homogeneous and isotropic spatial geometries: the Euclidean space $\mathbf{E}^3$, the hyperbolic space $\mathbf{H}^3$, the real projective space $\mathbf{RP}^3$, and the sphere $S^3$. In any of those cases, the universe would look the same all around us and be independent of our position. Thus, the sphere is possible but not necessary. The first two of those cases correspond to the "flat" and "open" Big Bang cosmologies, respectively, and the last two are variations of the "closed" cosmology.

And if we don't require the universe to be completely homogeneous and isotropic, but merely look like it in the part of it we observe, then much more exotic geometries are also possible. With cosmological inflation, virtually any large-scale geometry can be "blown up" to look flat, homogenous, and isotropic within our observable portion--even though it might not be s beyond the our cosmological horizon.

Otherwise, an instantaneous infinite Euclidean space model would work.

It does work, in the sense of being consistent with observations. In fact, in the standard ΛCDM cosmological model, space is infinite and Euclidean.

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