Exercise 2 — Cartesian Products

Problem

Give the geometric interpretation of the following Cartesian products:

(a) two line segments; (b) two lines; (c) a line and a circle; (d) a line and a disk; (e) two circles; (f) a circle and a disk.

Solution

Let $I = [0,1]$, $\mathbb{R}$ be the real line, $S^1 = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ be the unit circle, and $D^2 = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 1 \}$ be the closed unit disk.

Product	Notation	Geometric interpretation
(a) $I \times I$	Two segments	Rectangle (including interior)
(b) $\mathbb{R} \times \mathbb{R}$	Two lines	Plane $\mathbb{R}^2$
(c) $\mathbb{R} \times S^1$	Line $\times$ circle	Infinite cylinder (without caps)
(d) $\mathbb{R} \times D^2$	Line $\times$ disk	Solid infinite cylinder
(e) $S^1 \times S^1$	Circle $\times$ circle	Torus (surface of a doughnut)
(f) $S^1 \times D^2$	Circle $\times$ disk	Solid torus (filled doughnut)

Remarks

The general principle: the Cartesian product $X \times Y$ of two sets that are “geometric objects” of dimensions $p$ and $q$ respectively yields an object of dimension $p + q$ — provided one interprets “dimension” appropriately.

• (a) $1 + 1 = 2$ (rectangle)
• (c) $1 + 1 = 2$ (cylinder surface, a 2-manifold)
• (d) $1 + 2 = 3$ (solid cylinder)
• (e) $1 + 1 = 2$ (torus surface)
• (f) $1 + 2 = 3$ (solid torus)