Exercise 5, § 4

Let B be the Alexandroff compactification of the space $\mathbf{N}\times ]0,1[$ and let $o$ be its point at infinity (“Hawaiian earring”).

a) Prove that the space B is connected, locally connected, and locally connected by arcs.

b) Let $(r_n)_{n\in\mathbf{N}}$ be a strictly decreasing sequence of real numbers tending to 0. For every $n\in \mathbf{N}$, denote by $C_n$ the circle of the numerical plane with center $(r_n,0)$ and radius $r_n$; let P be the union of the $C_n$. Prove that B is homeomorphic to P and that every homeomorphism of B onto P maps the point $o$ onto the origin of the plane.

c) For $n\in \mathbf{N}$, denote by $P_n= C_0\cup \cdots \cup C_n$ and $f_n$ the unique mapping of P into $P_n$ such that $f_n(x) =x$ for $x\in P_n$ and $f_n(x) = 0$ otherwise. Prove that $f_n$ is continuous.

Let E and $E'$ be coverings of P. Prove that the canonical mapping Mor$_P(E',E)\rightarrow$ lim$\longrightarrow_n$ Mor$_P(E'_{P_n},E_{P_n})$ is a bijection.

Let E be a covering of P; prove that there exists $n\in \mathbf{N}$ such that E is isomorphic to $f_m^*(E_{P_m})$, for every integer $m\geqslant n$.