Is restriction of a projection to a connected subset a quotient map?

Question

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ be connected. Is $f \restriction C = f|(C \to f[C])$ a quotient map?

Answer 1

No! Consider $X=S^1=\mathbb{R}/\mathbb{Z}$, $Y=[0, 1]$, and $C$ the image of $[0, 1) \to S^1 \times [0, 1]$, $t \mapsto ([t], t)$. Informally, $Z=S^1 \times [0, 1]$ is a cylinder, and the set $C$ is a curve which goes from the top to the bottom in one turn.

Note that $f|_C\colon C \to f(C)$ is bijective but not an homeomorphism, as $C$ is not compact, while $f(C)=S^1$ is compact. Therefore, $f|_C$ is not a quotient map.