Help with gluing together surfaces of infinite genus

Question

Let $S=\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points from, say, $\mathbb{C}$: $S=\mathbb{C}\backslash\left\{ z_{m}:m\in\mathbb{Z}\right\}$. Let $\nu$ be an integer $\geq2$, and suppose we have $\nu$ copies of $S$ (written $S_{1},\ldots,S_{\nu}$) to be glued together along the “holes” (ex: for each $m$, the surfaces $S_{1},\ldots,S_{\nu}$ get glued together at the hole at $z_{m}$). I'm pretty sure that the resultant topological space (call it $T$) is then: $$T=\prod_{n=1}^{\nu}S=\prod_{n=1}^{\nu}\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$$ that is, $T$ is the direct product of $\nu$ copies of $S$. Is there a simpler description of $T$ (i.e., is there a way to interchange $\prod$ and $\bigwedge$; is there a way to write things in terms of $\mathbb{T}^{k}$ for some $k$s (where $\mathbb{T}^{k}=\prod_{i=1}^{k}\mathbb{S}^{1}$), etc.)?