Section Three Examples
In this section we consider three specific examples of a product of topological spaces.
Activity 20.3.
Let \(X = [1,2]\) and \(Y = [3,4]\) as subspaces of \(\R^2\text{.}\)
(a)
Explain in detail what the product space \(X \times Y\) looks like.
(b)
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 \(Y\text{.}\)
Activity 20.4.
Let \(X = \R\) and \(Y = S^1 = \{(x,y) \mid x^2 + y^2 = 1\}\text{,}\) the unit circle as a subset of \(\R^2\text{.}\)
(a)
Draw a picture of \(\R\text{.}\) For each \(x \in \R\text{,}\) the set \(\R_x = \{(x, y) \mid y \in S^1\}\) is a subset of \(\R \times S^1\text{.}\) On your graph of \(\R\text{,}\) draw pictures of \(\R_x\) for \(x\) equal to \(-1\text{,}\) \(0\text{,}\) and \(1\text{.}\) Explain in detail what the product space \(\R \times S^1\) looks like.
(b)
Consider the sets of the form \(B \cap S^1\text{,}\) where \(B\) is an open ball in \(\R^2\) (relatively open sets in \(S^1\)). What do these sets look like?
(c)
Describe the shape of the basis elements for the product topology on \(\R \times S^1\) that result from products of the form \(U \times V\text{,}\) where \(U\) is an open interval in \(\R\) and \(V\) is the intersections of \(S^1\) with an open ball in \(\R^2\text{.}\)
Activity 20.5.
Let \(2S^1 = \{(x,y) \mid x^2 + y^2 = 4\}\) be the circle of radius \(2\) centered at the origin as a subset of \(\R^2\text{.}\) In this activity we investigate the space \(2S^1 \times S^1\text{.}\)
(a)
Draw a picture of \(2S^1\) in the \(xy\)-plane. For each \(p \in S^1\text{,}\) the set \(S^1_p = \{(p, y) \mid y \in S^1\}\) is a subset of \(S^1 \times S^1\text{.}\) On your graph of \(S^1\text{,}\) draw pictures of \(S^1_p\) for \(p\) equal to \((1,0)\text{,}\) \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\text{,}\) 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.
(b)
Consider the sets of the form \(B \cap S^1\text{,}\) where \(B\) is an open ball in \(\R^2\text{.}\) What do these sets look like?
(c)
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 \(\R^2\text{.}\)
