Relation between Homogeneity and Isotropy of space? As per my understanding so far, homogeneity of space doesn't require a special vantage point (all points in space are "equivalent" to each other) and is a universal statement in that sense; whereas for the isotropy of space, we talk with respect to a single point (the points on any circle, centered at this point, are all "equivalent" to each other)
Clearly then this implies that the isotropy trivially follows from homogeneity of space. But then whenever we are talking of symmetries we always talk of them separately as if they are independent of each other. For example, I have always seen these assumptions stated separately when we define inertial frames. Moreover (continuing this example), in the derivations of lorentz transformation equation, homogeneity assumption is used to argue for linearity of transformation equations and isotropy is used to further eliminate certain coefficients (made possible by that symmetry).
Indeed, the way I saw these concepts being used always made me feel that these concepts are quite different in flavor and independent in this sense but again, when I return back to think of these in isolation, what I said earlier is what comes to my mind, clearly in conflict with how these things are being used.
I reckon, there is clearly something wrong with my intuitive understanding of these basic concepts. Seeking clarification
P.S. Also, I saw a claim somewhere that isotropy about 2 points implies that space is homogeneous and again, I can't see the reason. What I understand is that the points on the circle around a point (isotropy assumed) does not form an equivalence class in that if for one of the points on the circle it is known that space is isotropic, we can't really conclude the same about the others (on that circle) that same must hold true just because those points were "equivalent".
Am I missing something?
 A: You are right about homogeneity. It means space is the same everywhere.
But isotropy means all directions are the same. This is a different concept.
Suppose you are in infinite space filled with a uniform electric field in the X direction. This is homogeneous, but not isotropic.
If you move to another point, you will still see a uniform field in the X direction extending to infinity. But if you rotate $180^o$, the electric field looks different.
A: It is most useful to understand these concepts with respect to what they imply mathematically, and then deduce their physical implications.
Homegeneity implies that the coordinate-dependent part of the metric that describes your space is only time-dependent. If, for example, you use a FRW model:
\begin{equation}
ds^{2} = -N^{2}\, dt^{2} + a^{2}\, dx^{i} dx^{i}
\end{equation}
then the lapse function $N$ and the scale factor $a$ can only be functions of time.
Isotropy on the other hand implies that there is no preferred spatial coordinate i.e. they all behave the same. What this further implies is that there is a complete rotational symmetry $O(n)$ (where $n$ the number of spatial coordinates for a $n+1$-dimensional spacetime), hence your manifold behaves like a sphere. Thus you can express the spatial coordinates as spherical ones:
\begin{equation}
dx^{i} dx^{i} = d\Omega ^{2} _{n}
\end{equation}
where $d\Omega ^{2} _{n}$ the $S^{n}$ metric, which is the metric of a unit $n$-sphere.
Now that we have a concrete mathematical explanation, the physical implications fall right out:

*

*Homogeneity tells you that there are no local particularities at any space on the manifold and everything is uniform everywhere.

*Isotropy expresses a symmetry of space which is the rotational one. The Universe should mathematically look the same (its various quantities remain invariant) after you execute a rotation towards any direction or series of such.

