Exercise 2, § 1

Let Y be a noncompact locally compact space that is countable at infinity and let $(Y_n)$ be an increasing sequence of compact subsets of Y forming a covering of Y and such that $Y_n \subset \overset{\circ}{Y}_{n+1}$ for all n, so that every compact subset of Y is contained in one of the $Y_n$ (GT, I, §9, No. 9, Cor. 1 of Prop. 15). Let $\beta Y$ be the Stone–Čech compactification of Y (GT, IX, §1, Exer. 7), so that the space $\mathscr{C}(\beta Y; \mathbf{R})$ may be identified with the space of bounded continuous real-valued functions on Y, and Y is a dense open set in $\beta Y$; for every function $f \in \mathscr{C}(\beta Y; \mathbf{R})$, Supp(f) is thus the closure in $\beta Y$ of the set of $y \in Y$ such that $f(y) \neq 0$. Recall that if $Z_1, Z_2$ are two subsets of Y, closed in Y and disjoint, then their closures $\overline{Z}_1, \overline{Z}_2$ in $\beta Y$ are also disjoint (GT, IX, §4, Exer. 17).

Let $\omega$ be a point of $\beta Y - Y$ and let $X = \beta Y - \{ \omega \}$. Show that, on the space $\mathscr{K}(X; \mathbf{R})$, the direct limit topology $\mathscr{T}$ defined in No. 1 is identical to the topology of uniform convergence. (If p is a semi-norm that is continuous for $\mathscr{T}$, show, by means of a contradiction argument, that there exists a number $c > 0$ such that $p(f) \leq c \|f\|$ for every function $f \in \mathscr{K}(X; \mathbf{R})$. In the contrary case, there would exist a sequence $(f_n)$ of functions in $\mathscr{K}(X; \mathbf{R})$ such that $\|f_n\| \leq 1 , \ p(f_n) \geq n , \ \mathrm{Supp}(f_n) \cap Y_n = \varnothing$, and

for $k < n$. Then let $Z_i \ (i = 0, 1)$ be the union of the sets $Y \cap \mathrm{Supp}(f_{2n+i})$; show that one of the two sets $Z_0, Z_1$ is relatively compact in X (GT, X, §4, Exer. 7), and from this deduce a contradiction.) From this, deduce that the space $\mathscr{K}(X; \mathbf{R})$ equipped with the topology $\mathscr{T}$ is not quasi-complete.