##SU(2)## homeomorphic with ##\mathbb S^3##

[cianfa72]

All that is needed is the decision of how you want to look at it and where to start from. We run around in circles here, partly because of the words "induced" or "subspace topology" which have been used without mentioning "induced by what and how?" or "subspace of what?".

See this link (2nd answer). The set of matrices $M(z,w)$ without constraint $|z|^2 + |w|^2 =1$ is homeomorphic with $\mathbb C^2 \cong \mathbb R^4$. The condition $|z|^2 + |w|^2 =1$ defines a 3-sphere $\mathbb S^3$ in $\mathbb R ^4$. Then restricting the above homeomorphism on $\mathbb S^3$ endowed with the subspace topology from $\mathbb R ^4$ we get the homeomorphism with $SU(2)$.

 

[fresh_42]

I will not respond to an again imprecisely phrased statement since $M(z,w)$ is undefined, there is no homeomorphism other than the identity, and a point on $\mathbb{S}^3$ has nothing to do with a point on $SU(2)$ without explaining the correspondence. I can understand it, nevertheless, it is sloppy. IMO, you have to learn to be way more precise than you usually are. Half of your questions resolve themselves once they are stated with the necessary precision. That does NOT mean to write more, au contraire, it is very often less!

This thread is closed.