Math Problems Vault

Problem V.6.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

If $x \in X$ and $\delta_x(f) = f(x)$ for all $f$ in $C_b(X)$ , show that $\left\| \delta_x\right\| = 1$ .

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Problem V.6.2 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

Prove that a subset of a completely regular space is completely regular.

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Problem V.6.3 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

(a)(b)(c)

Fill in the details of the proof of Theorem 6.2 (Stone-Cech Compactification): if $X$ is completely regular, then there is a compact space $\beta X$ such that

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(a)

there is a continuous map $\Delta : X \to \beta X$ with the property that $\Delta : X \to \Delta (X)$ is a homeomorphism;

(b)

$\Delta (X)$ is dense in $\beta X$ ;

(c)

if $f \in C_b(X)$ , then there is a continuous map $f^\beta : \beta X \to \mathbb {F}$ such that $f^\beta \circ \Delta = f$ .

Moreover, if $\Omega$ is a compact space having these properties, then $\Omega$ is homeomorphic to $\beta X$ .

Problem V.6.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

If $X$ is completely regular, $\Omega$ is compact, and $f : X \to \Omega$ is continuous, show that there is a continuous map $f^\beta : \beta X \to \Omega$ such that $f^\beta | X = f$ .

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Problem V.6.5 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

If $X$ is completely regular, show that $X$ is open in $\beta X$ if and only if $X$ is locally compact.

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Problem V.6.6 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

(a)(b)

Let $\mathbb {N}$ have the discrete topology. Let $\left\{ r_n : n \in \mathbb {N} \right\}$ be an enumeration of the rational numbers in $[0,1]$ . Let $S =$ the irrational numbers in $[0,1]$ and for each $s$ in $S$ let $\left\{ r_n : n \in N_s \right\}$ be a subsequence of $\left\{ r_n \right\}$ such that $s = \lim \left\{ r_n : n \in N_s \right\}$ . Show:

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(a)

if $s, t \in S$ and $s \neq t$ , $N_s \cap N_t$ is finite;

(b)

if for each $s$ in $S$ , $\cl N_s =$ the closure of $N_s$ in $\beta \mathbb {N}$ and $A_s = (\cl N_s) \setminus \mathbb {N}$ , then $\left\{ A_s : s \in S \right\}$ are pairwise disjoint subsets of $\beta \mathbb {N} \setminus \mathbb {N}$ that are both open and closed.

Problem V.6.7 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

Show that if $X$ is normal, $\tau \in \beta X$ , and there is a sequence $\left\{ x_n \right\}$ in $X$ such that $x_n \to \tau$ in $\beta X$ , then $\tau \in X$ . If $X$ is not normal, is the result still true?

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Problem V.6.8 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.6 "An Application: The Stone-\v{C}ech Compactification"

Let $X$ be the space of all ordinals less than the first uncountable ordinal and give $X$ the order topology. Show that $\beta X =$ the one point compactification of $X$ . (You can find the pertinent definitions in Kelley [1955].)

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Problem VIII.2.7 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VIII.2 "Abelian C*-Algebras and the Functional Calculus in C*-Algebras"

If $X$ is completely regular, then $C_b(X)$ is a $C^*$ -algebra and its maximal ideal space is the Stone--Cech compactification of $X$ .

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Completely regular spaces and the Stone-Cech compactification

References