Juts a quick question. In Freitag's Complex Analysis as an example for The Quotient Topology it comes:
The "modular space" $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z}).$
Every element in $\mathbb{H}$ can be mapped by a linear fractional transformation in $\mathbb{H}/ \mathrm{SL}_2(\mathbb{Z})$ to some fixed element ${\{\tau_0}\}$ in $\mathbb{H}$ so is it true to say $\mathbb{H}/ \mathrm{SL}_2(\mathbb{Z}) \cong {\{\tau_0}\}$? So basically a modular space is just any single point in $\mathbb{H}$?
I have a little background in Modular Forms so much appreciated a simple explanation.
While all elements of $\mathbb{H}$ are $\mathrm{SL}_2(\mathbb{R})$-equivalent, this is no longer the case if $\mathbb{R}$ is replaced by $\mathbb{Z}$. If (as you say) you have some background in modular forms, you've likely seen a picture of the standard fundamental domain for the modular group: $$ \mathcal{F}=\{z=x+iy:y>0,-\frac{1}{2}\leq x\leq \frac{1}{2},|z|\geq 1\}$$ You can visualize the quotient space by gluing together the boundary of $\mathcal{F}$: the points $-\frac{1}{2}+y$ and $\frac{1}{2}+y$ are identified, as are $z$ and $-\frac{1}{z}$ if $z\in\mathcal{F},|z|=1$.