(1.37) Let $X$ be the plane $\mathbf{R}^2$ equipped with the metric topology and let $Y$ be a curve in $X$. Take an open ball in $X$ and intersect it with $Y$: you might get something like an open interval in the curve. This is then open in the subspace topology on $Y$.

(2.34) Let $X$ be the plane $\mathbf{R}^2$ again and let $Y=[0,1]\times[0,1]$ be the closed square in $X$. In the subspace topology on $Y$, the subset $Y=[0,1]\times[0,1]\subset Y$ is open (even though it's closed as a subset of $X$). Of course, this must be true: otherwise the subspace topology would not satisfy the axioms of a topology. To see in this case why $Y$ is open, take an open ball $V=B_0(2)$ of radius 2 centred at $0$ in $X$: we have $Y=V\cap Y$, so $Y$ is open in the subspace topology.

(4.28) The circle $S^1=\{z\in\mathbf{C}\ :\ |z|=1\}$ is a subset of the complex plane so it inherits a subspace topology.

(5.20) The $n$-sphere is the subset $\{(x_0,\ldots,x_n)\in\mathbf{R}^{n+1}\ :\ \sum_{k=0}^nx_k^2=1\}$ inherits a subspace topology from $\mathbf{R}^{n+1}$

(6.07) The torus $T^2$ is a subset of $\mathbf{R}^3$ so it inherits a topology. Note that I am only talking about the surface of the torus (with longitude and latitude coordinates) so this is a 2-dimensional space. More generally, any closed orientable surface (genus 2, genus 3,...) can be embedded in $\mathbf{R}^3$ so inherits a topology.

(7.08) The torus can also be embedded in $\mathbf{R}^4$. It inherits another topology from this embedding, but this topology is homeomorphic to the topology it gets from $\mathbf{R}^3$ (isomorphic in the category of topological spaces). To see how the torus embeds in $\mathbf{R}^4$, let $(\theta,\phi)$ be longitude/latitude coordinates on the torus and embed this point as $(\cos\theta,\sin\theta,\cos\phi,\sin\phi)\in\mathbf{R}^4$. Although the torus in $\mathbf{R}^3$ and the torus in $\mathbf{R}^4$ are homeomorphic (the same topological space) the geometries they inherit from their ambient spaces are very different (in $\mathbf{R}^3$ the torus has Gaussian curvature which varies from point to point (from positive to negative); in $\mathbf{R}^4$ the Gaussian curvature of the torus is zero).

The product topology on $Y_1\times Y_2$ is the coarsest topology on $Y_1\times Y_2$ making both projection maps $p_k\colon Y_1\times Y_2\to Y_k$, $p_k(y_1,y_2)=y_k$, continuous.