Meaning of "homeomorphic with the group $O_3^+(2)$"

Question

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?

Answer 1

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.

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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.