Exercise 10, § 5

For every integer $n\geqslant$ 1, let $C_n$ be the circle of the numerical plane with center $(1/n,0)$ and radius $1/n$; let P be the union of the spaces $C_n$, for $n\geqslant 1$. For every integer $n\geqslant 1$, let $B_n$ be the set of points $(x, y)$ of the plane such that $(nx-1)^2+n^2y^2\leqslant 1\leqslant ((n+1)x-1)^2+(n+1)^2y^2$; let B be the set of points $(x, y)$ of the plane such that $(x-1)^2+y^2\leqslant 1$. Put $B_0= P$ and let A be the quotient space of the sum space of the family $(B_n)_{n\geqslant 0}$ by the least fine equivalence relation which identifies $(b, n)$ and $(b,0)$ if $b\in C_n\cup C_{n+1}$.

(a) Prove that there exists a unique continuous mapping $p: A\rightarrow B$ which, for every integer $n\geqslant 0$ and every $b\in B_n$, associates the point $b$ with the class of $(b, n)$. Prove that $p$ is bijective but that it is not a homeomorphism. (b) Prove that the space A is locally arcwise connected.

(c) Let $a$ be the class of the point $(0,0)$. Prove that the quotient topology of the compact convergence topology on $\pi_1(A, a)$ is the coarse topology. (d) Prove that the space A is simply connected.

(e) Let $j$ be the canonical mapping of $C = B_0$ into A. Prove that the mapping $j_*:\pi_1(C,0)\rightarrow \pi_1(A, j(0))$ is surjective and that its kernel has the power of the continuum.

(f) Prove that the group $\pi_1(A, j(0))$ has the power of the continuum. In particular, the space A is not unlaceable.