Exercise 7, § 4

Let X be a completely regular non-compact space.

a) Now prove that the following properties are equivalent:

α) The subalgebra of $\mathscr{C}^b(X; \mathbf{R})$ formed by the functions of the form $c + f$, where c is a constant and $f$ has a compact support (IX, p. 46), is dense in $\mathscr{C}^b(X; \mathbf{R})$.

β) If $\beta X$ is the Stone-Čech compactification of X, $\beta X - X$ is reduced to a single point.

γ) There exists only one precompact uniform structure of space compatible with the topology of X.

δ) If A, B are two closed subsets of X, normally separated (IX, p. 87, exerc. 15), then one of the two sets A, B is compact.

ζ) For every uniform structure $\mathscr{U}$ on X compatible with the topology of X and every entourage V of $\mathscr{U}$, there exists a set L, complementary to a compact subset of X, such that $L \times L \subset V$.

θ) There exists only one uniform structure compatible with the topology of X.

(Observe that α) implies that X is locally compact, hence open in βX, and deduce that α) implies β); to see that β) implies α), use the Weierstrass-Stone theorem. To show that β) implies γ), use the fact that the uniform structure induced on X by that of βX is the finest of the uniform structures of precompact space compatible with the topology of X. To prove that γ) implies δ), use exerc. 15 of IX, p. 87. To see that δ) implies ζ), first show that for every entourage V of U, there exists z ∈ X such that V(z) is not compact; otherwise, X would be complete for U and paracompact (II, p. 37, exerc. 9), hence normal; show then with the aid of δ) that X would be semi-compact (IX, p. 93, exerc. 13) and conclude with the aid of II, p. 37, exerc. 6. Apply this result then to an entourage V defined by an equation f(x, x') ≤ 1 for a continuous distance f on X, and deduce from δ) that there exists a set L complementary to a compact subset of X and such that L × L ⊂ V. Finally, to establish that ζ) implies θ), observe that every ultrafilter on X is either convergent, or finer than the filter of the complements of the relatively compact subsets of X.)

b) Now prove that a space X possessing the equivalent properties of a) is weierstrassian (IX, p. 88, exerc. 20). (Use exerc. 17 of IX, p. 87, or observe that if X is not weierstrassian, there exists a sequence (x_n) of points of X without a point of adherence, and a continuous mapping f of X into {0, 1} such that f(x_{2n}) = 0 and f(x_{2n+1}) = 1).

c) Let Y be a completely regular non-compact space. Now prove that the subspace X of βY, complementary to a point of βY − Y, satisfies the conditions of a).