I am trying to understand the statement at http://mathworld.wolfram.com/SpecialUnitaryGroup.html

$SU(2)$ is homeomorphic with the orthogonal group $O_3^+(2)$.

First, for groups isn't the term homeomorphic equivalent to isomorphic?
Or are they trying to say something weaker here, that there exists an invertible map between these continuous groups which preserves a sense of the 'neighborhoods' of group elements, but not necessarily the actual group structure!? But if the map doesn't preserve the group structure, I struggle to see what such a map is useful for, so I hope it is okay to read that as isomorphic.

Second, what is denoted by $O_3^+(2)$?
I have not seen that notation before and am having difficulty finding the right name to find it in a search.
I believe the superscript + can refer to the component having the identity, for instance the "restricted Lorentz group $SO^+(1,3)$".

So if I understand correctly, the elements of $O^+(2)$ could be parameterized with one real term $$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$ where as SU(2), being isomorphic with the unit quaternions, could be parameterized with three real terms. So maybe somehow the subscript three refers to three copies of $O^+(2)$ connected somehow?

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    $\begingroup$ I would assume that they mean precisely what they write. Being homeomorphic is completely unrelated to the group structure ans simply means that they have "the same" topological structure (not sure what you mean by continuous groups. Did you mean topological groups?) $\endgroup$ – Tobias Kildetoft Apr 6 '17 at 17:55
  • $\begingroup$ @TobiasKildetoft By "continuous groups" I meant groups who elements could be specified by continuous parameters. Choosing such a parameterization would then induce a topology I guess, but I was not considering this fundamental to the group itself. Even if we consider this fundamentally part of the object in question, I would still expect (apparently incorrectly?) that a morphism between these groups would preserve the group structure. If they are throwing out the group structure, why map to another group, instead of just a topological space? $\endgroup$ – PPenguin Apr 6 '17 at 18:22
  • $\begingroup$ Explaining what $O_3^+(2)$ means would clear up a lot here. $\endgroup$ – PPenguin Apr 6 '17 at 18:24
  • $\begingroup$ Link to original question where notation came up: math.stackexchange.com/questions/2218186/… $\endgroup$ – PPenguin Nov 3 '18 at 22:39

Short answer, that sentence from Mathworld is almost all typos:

  1. It appears "homeomorphic" is a typo on that page. (should read as "homomorphic")

  2. The term $O^+_3(2)$ appears to also be a typo for $O^+_3$ which is the notation used for $SO(3)$ in the cited reference.

On that Mathworld page Eric Weisstein cites as a reference "Mathematical Methods for Physicists, Third Edition", which if you search you can find some sections online. Here are some snippets, which given this is what was cited, is why I believe the Mathworld statement has multiple typos. (emphasis as in original)

On pg 255, in the section titled "$SU(2) - O^+_3$ Homomorphism"

Our real orthogonal group $O^+_3$, determinant +1, clearly describes rotations in ordinary three-dimensional space with the important characteristic of leaving $x^2 + y^2 + z^2$ invariant.

pg 257

The correspondance is 2 to 1, or $SU(2)$ and $O^+_3$ are homomorphic.


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