Consider the horizontal strip $ \ A=\{(x,y): \ x \in \mathbb{R} \ \ and \ \ -1 \leq y \leq 1 \} \ $.
Define a relation $ \ \sim \ $ on $ \ A \ $ by $ \ (a,b) \sim (c,d) \ $ iff $ \ a-c \in \mathbb{Z} \ $ and $ \ d=(-1)^{a-c} c \ $ .
This relation is an $ \ equivalence \ \ relation \ $
Then answer the following questions:
$ \ (i) \ $ Given an element $ \ (a,b) \in A \ $ describe its equivalence class $ \ [(a,b)] \ $ geometrically
$ (ii) \ $ What is $ \ A / \sim \ $ topologically ?
Answer:
$ (i) \ $
Since $ a-c \in \mathbb{Z} \ $ , we conclude that
$ d=0 \ \ or \ \ d=c \ \ or \ \ d=-c \ $
Thus $ [(a,b)]=\{(c,0), (c,c) , (c,-c) \} \ $
Further $ -1 \leq d \leq 1 \ $
Thus the equivalence class contains the sets $ \ \{(x,y): -1 \leq x \leq 1 , \ y=0 \} , \ \{(x,y): -1 \leq x \leq 1 ,\ y=x \} , \ \{(x,y): -1 \leq x \leq y , \ y=-x\} \ $
Thus geometrically $ \ [(a,b)] \ $ represents the following graph
$ (ii) \ $
I think $ \ A / \sim \ $ is a quotient space .
But I can not describe it.
Help me out in the both questions.