Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\mathbb{C}}$. Since $SU(2)$ is simply-connected $\exp: \mathfrak{su}(2) \rightarrow SU(2)$ covers the whole lie group, thus the collection $\{e^{X}: X\in\mathfrak{su}(2)\}$ is homeomorphic to $S^3$. Now consider writing
$$ X = \alpha_1\frac{i\sigma_1}{2} + \left(\alpha_2\frac{i\sigma_2}{2} + \alpha_3\frac{i\sigma_2}{2}\right),\ \ \alpha_i \in \mathbb{C}.$$
and computing $$e^{\alpha_1\frac{i\sigma_1}{2}}e^{\alpha_2\frac{i\sigma_2}{2} + \alpha_3\frac{i\sigma_2}{2}}.$$
Gilmore's book on Lie groups and Lie algebras for physicists and engineers appears to assign split products of this form topologies corresponding to the parts, e.g. this one would be $U(1)\times SU(2)/U(1)$ or equivalently $\mathbb{T}^1\times S^3/\mathbb{T}^1$. What is the logic / deeper theory behind assigning this topology to the resulting manifold? In particular, how does one show the second term has that division?