Let $T=S^1 \times S^1$ be the torus with meridian $A=S^1 \times \{1\} \subset T$
Let $C=S^1 \times [-1, 1]$ be the cylinder with base circles $B=S^1 \times \{-1, 1\}$.
Show that $C/B \approx T/A$
So I want to show that the quotient space are homeomorphic.
I know that if we let $q : X \rightarrow X/~$ be a quotient map $f:X \rightarrow Y$ be a continuous surjection to a Haussdorff space such that $q(a)=q(b)$ if and only if $f(a)=f(b)$ for all $a, b \in X$. IF $X$ is compact then $X/~$ is compact then $X/~ \approx Y$.
I was thinking of using this fact to start by considering a surjection $f:C \rightarrow T/A$ and consider what points map to $C/B$
Could someone help me find a way through this problem?