Exercise 19, § 11

Let X be a locally compact space. It can be shown that X can be identified with a dense open subspace of the universal compact space $\tilde{X}$ constructed in § 9, Exercise 25 (cf. Chapter IX, § 1, Exercise 8). Suppose also that X is connected and locally connected.

a) Show that, for every compact subset K of X, the set of components of $X - K$ which are not relatively compact is finite [cf. Exercise 14 b)]. Deduce that a point of $\tilde{X} - X$ belongs to the closure of at most one of the components of $X - K$ [if $A_i$ ($1 \leq i \leq n$) are the non-compact components of $X - K$, and $C_i$ the compact set of points of $\tilde{X} - X$ which lie in the closure of $A_i$, construct a compact space whose underlying set is the sum of X and the sets $C_i$, such that X is identified with a dense open subspace of this space].

b) Let $R' \{ x, y \}$ be the following equivalence relation on $\tilde{X} - X$: "for every compact subset K of X, x and y lie in the closure of the same non-compact component of $X - K$". Let R be the equivalence relation on $\tilde{X}$ whose equivalence classes are the classes mod $R'$ and the subsets consisting of a single point of X. Show that R is Hausdorff and that X may be identified with a dense open subspace of the compact space $X^b = \tilde{X}/R$; show that in the compact space $X^b - X$ every point is the intersection of its neighbourhoods which are both open and closed. The points of $X^b - X$ are called the ends of the space X.

c) Let $Y$ be a compact space such that $X$ may be identified with a dense open subspace of $Y$ and such that in the compact space $Y - X$ every point is the intersection of its neighbourhoods which are both open and closed (*). Show that there is a continuous mapping of $X^b$ into $Y$ which induces the identity mapping on $X$. [If $Y - X$ is the union of two disjoint non-empty closed sets $A$ and $B$, show that there exist two disjoint open subsets $U, Y$ in $V$ such that $A \subset U, B \subset V$ and such that $Y - (U \cup V)$ is compact].

d) Show that if $X$ is $\sigma$-compact, then $X^b - X$ has a countable base.

e) The real line has two ends, and $\mathbf{R}^n$ ($n \geq 2$) has one. Give an example of a connected, locally connected, locally compact subspace of $\mathbf{R}^2$ for which the set of ends has the power of the continuum.*