Exercise 15, § 3

Consider the following three subspaces of $\mathbf{R}$: $X_1 = ] - 1, 1[$, $X_2 = ] - 2, - 1[, X_3 = ] 1, 2[$. Let $X$ be their union; $X$ is then the sum of these three subspaces. Let $A_2$ (resp. $A_3$) be the set of irrational numbers contained in $X_2$ (resp. $X_3$); let $B_1$ (resp. $B_2$) be the set of rational numbers contained in ]—$1, 0[$ (resp. $X_2$); and let $C_1$ (resp. $C_3$) be the set of rational numbers contained in ]$0, 1[$ (resp. $X_3$). Let $X/R$ be the space obtained by pasting together the $X_i$ (i) along $A_2$ and $A_3$ by means of the homeomorphism $x \to x + 3$; (ii) along $B_1$ and $B_2$ by means of the homeomorphism $x \to x - 1$; (iii) along $C_1$ and $C_3$ by means of the homeomorphism $x \to x + 1$. Show that the canonical image of $X_1$ in $X/R$ is not homeomorphic to $X_1$. If $Y = X_1 \cup B_2 \cup C_3$ is the saturation of $X_1$ with respect to $R$, show that the quotient space $Y/R_Y$ is not homeomorphic to the canonical image of $Y$ in $X/R$. \*