Exercise 17, § 11

Let $I_0$ be the interval $[ -1, 1 ]$ of $\mathbf{R}$ and let $P_0$ be the quotient space of $I_0$ obtained by identifying the two points $1/2$ and $1$; let $B_0$ be the quotient space of $I_0$ obtained by identifying the points $1/2$ and $1$ and also identifying $-1/2$ and $-1$. Let $I$ be the open interval ]— i, i[ in $\mathbf{R}$, and let P (resp. B) be the canonical image of I in $P_0$ (resp. $B_0$).

a) Show that no two of the three spaces I, P, B are homeomorphic (to show that I and P are not homeomorphic, notice that there are points in P whose complement is connected; similarly for the other pairs of spaces).

b) Let X be the sum of a countably infinite family of spaces homeomorphic to I and a countably infinite family of spaces homeomorphic to B. Let Y be the sum of X and P. Show that X and Y are not homeomorphic, but that (i) there is a continuous bijection of X onto Y and a continuous bijection of Y onto X; (ii) X is homeomorphic to an open subspace of Y and Y is homeomorphic to an open subspace of X.*