Topology: An Inquiry-Based Approach

Explain in detail what the product space $X\times Y$ looks like.

Find, if possible, an open subset of $X\times Y$ that is not of the form $U\times V$ where $U$ is open in $X$ and $V$ is open in $Y\text{.}$

Draw a picture of $\mathbb{R}\text{.}$ For each $x\in \mathbb{R}\text{,}$ the set ${\mathbb{R}}_x=\{(x,y)\mid y\in S^1\}$ is a subset of $\mathbb{R}\times S^1\text{.}$ On your graph of $\mathbb{R}\text{,}$ draw pictures of ${\mathbb{R}}_x$ for $x$ equal to $-1\text{,}$ $0\text{,}$ and $1\text{.}$ Explain in detail what the product space $\mathbb{R}\times S^1$ looks like.

Consider the sets of the form $B\cap S^1\text{,}$ where $B$ is an open ball in ${\mathbb{R}}^2$ (relatively open sets in $S^1$). What do these sets look like?

Describe the shape of the basis elements for the product topology on $\mathbb{R}\times S^1$ that result from products of the form $U\times V\text{,}$ where $U$ is an open interval in $\mathbb{R}$ and $V$ is the intersections of $S^1$ with an open ball in ${\mathbb{R}}^2\text{.}$

Draw a picture of $2S^1$ in the $xy$-plane. For each $p\in S^1\text{,}$ the set $S_p^1=\{(p,y)\mid y\in S^1\}$ is a subset of $S^1\times S^1\text{.}$ On your graph of $S^1\text{,}$ draw pictures of $S_p^1$ for $p$ equal to $(1,0)\text{,}$ $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\text{,}$ and $(0,1)\text{.}$ Orient the graphs so that the copies of $S^1$ are perpendicular to $2S^1\text{.}$ Explain in detail what the product space $2S^1\times S^1$ looks like.

Consider the sets of the form $B\cap S^1\text{,}$ where $B$ is an open ball in ${\mathbb{R}}^2\text{.}$ What do these sets look like?

Describe the shape of the basis elements for the product topology on $2S^1\times S^1$ that result from products of the form $U\times V\text{,}$ where $U$ and $V$ are intersections of $S^1$ with open balls in ${\mathbb{R}}^2\text{.}$