What is continuous map?

Example 6.11

The quotient $\R/\Z$ (identify $x$ and $x + n$) is homeomorphic to the circle $S^1 = \{z \in \C : \abs z = 1\}$: the map $x \mapsto \eu^{2\iu\pi x}$ passes to a continuous bijection $\R/\Z \to S^1$ (Proposition 6.10(b)); its inverse is continuous by the compactness argument of Corollary 6.14 below (Exercise 6.5 details everything, including why $\R/\Z$ is Hausdorff and compact). Likewise $[0,1]$ with endpoints glued is $S^1$, the square with opposite sides glued is the torus, and gluing is finally a theorem, not a picture.

Example 6.22 (The topologist’s sine curve)

Let $\Gamma = \{(x, \sin\frac1x) : 0 < x \leq 1\}$ and $S = \bar\Gamma = \Gamma \cup (\{0\}\times[-1,1])$ (every point $(0, y)$, $\abs y \leq 1$, is a limit of points of $\Gamma$: solve $\sin\frac1x = y$ near $0$). Then $S$ is connected — closure of the connected $\Gamma$, a continuous image of $(0, 1]$ (Theorem 6.20(3)) — but not path-connected: a path from $(1, \sin 1)$ to $(0,0)$ would have to traverse abscissas $\to 0$ while the ordinate oscillates between $\pm1$; Exercise 6.9 makes this rigorous. Connectedness and path-connectedness genuinely differ.