
6.8.E: Problems on Baire Categories and Linear Maps


Exercise $$\PageIndex{1}$$

Verify the equivalence of the various formulations in Definition 1. Discuss: $$A$$ is nowhere dense iff it is not dense in any open set $$\neq \emptyset$$.

Exercise $$\PageIndex{2}$$

Verify Examples (a) to (e). Show that Cantor's set $$P$$ is uncountable.
[Hint: Each $$p \in P$$ corresponds to a "ternary fraction," $$p=\sum_{n=1}^{\infty} x_{n} / 3^{n},$$ also written $$0 . x_{1}, x_{2}, \ldots, x_{n}, \ldots,$$ where $$x_{n}=0$$ or $$x_{n}=2$$ according to whether $$p$$ is to the left, or to the right, of the nearest "removed" open interval of length $$1/ 3^{n}$$. Imitate the proof of Theorem 3 in Chapter 1, §9, for uncountability. See also Chapter 1, §9, Problem 2(ii).]

Exercise $$\PageIndex{3}$$

Complete the missing details in the proof of Theorems 1 to 4.

Exercise $$\PageIndex{4}$$

Prove the following.
(i) If $$B \subseteq A$$ and $$A$$ is nowhere dense or meagre, so is $$B$$.
(ii) If $$B \subseteq A$$ and $$B$$ is nonmeagre, so is $$A$$.
[Hint: Assume $$A$$ is meagre and use (i)).]
(iii) Any finite union of nowhere dense sets is nowhere dense. Disprove it for infinite unions.
(iv) Any countable union of meagre sets is meagre.

Exercise $$\PageIndex{5}$$

Prove that in a discrete space $$(S, \rho),$$ only $$\emptyset$$ is meagre.
[Hint: Use Problem 8 in Chapter 3, §17, Example 7 in Chapter 3, §12, and our present Theorem 1.]

Exercise $$\PageIndex{6}$$

Use Theorem 1 to give a new proof for the existence of irrationals in $$E^{1}$$.
[Hint: The rationals $$R$$ are a meagre set, while $$E^{1}$$ is not.]

Exercise $$\PageIndex{7}$$

What is wrong about this "proof" that every closed set $$F \neq \emptyset$$ in a complete space $$(S, \rho)$$ is residual: "By Theorem 5 of Chapter 3, §17, $$F$$ is complete as a subspace. Thus by Theorem 1, $$F$$ is residual." Give counterexamples!

Exercise $$\PageIndex{8}$$

We call $$K$$ a $$\mathcal{G}_{\delta}$$-set and write $$K \in \mathcal{G}_{\delta}$$ iff $$K=\bigcap_{n=1}^{\infty} G_{n}$$ for some open sets $$G_{n}.$$
(i) Prove that if $$K$$ is a $$\mathcal{G}_{\delta}$$-set, and if $$K$$ is dense in a complete metric space $$(S, \rho),$$ i.e., $$\overline{K}=S,$$ then $$K$$ is residual in $$S$$.
[Hint: Let $$F_{n}=-G_{n}.$$ Verify that $$(\forall n) G_{n}$$ is dense in $$S,$$ and $$F_{n}$$ is nowhere dense. Deduce that $$-K=-\bigcap G_{n}=\cup F_{n}$$ is meagre. Use Theorem 1.]
(ii) Infer that $$R$$ (the rationals) is not a $$\mathcal{G}_{\delta}$$-set in $$E^{1}$$ (cf. Example(c)).

Exercise $$\PageIndex{9}$$

Show that, in a complete metric space $$(S, \rho),$$ a meagre set $$A$$ cannot have interior points.
[Hint: Otherwise, $$A$$ would obtain a globe $$G.$$ Use Theorem 1 and Problem 4(ii).]

Exercise $$\PageIndex{10}$$

(i) A singleton $$\{p\} \subseteq(S, \rho)$$ is nowhere dense if $$S$$ clusters at $$p;$$ otherwise, it is nonmeagre in $$S$$ (being a globe, and not a union of nowhere dense sets).
(ii) If $$A \subseteq S$$ clusters at each $$p \in A,$$ any countable set $$B \subseteq A$$ is meagre in $$S$$.

Exercise $$\PageIndex{11}$$

(i) Show that if $$\emptyset \neq A \in \mathcal{G}_{\delta}$$ (see Problem 8) in a complete space $$(S, \rho),$$ and $$A$$ clusters at each $$p \in A,$$ then $$A$$ is uncountable.
(ii) Prove that any nonempty perfect set (Chapter 3, §14) in a complete space is uncountable.
(iii) How about $$R$$ (the rationals) in $$E^{1}$$ and in $$R$$ as a subspace of $$E^{1}?$$ What is wrong?
[Hints: (i) The subspace $$(\overline{A}, \rho)$$ is complete (why?); so $$A$$ is nonmeagre in $$\overline{A},$$ by Problem 8. Use Problem 10(ii). (ii) Use Footnote 3.]

Exercise $$\PageIndex{12}$$

If $$G$$ is open in $$(S, \rho),$$ then $$\overline{G}-G$$ is nowhere dense in $$S$$.
[Hint: $$\overline{G}-G=\overline{G} \cap(-G)$$ is closed; so
$\overline{(\overline{G}-G)^{0}}=(\overline{G}-G)^{0}=(\overline{G} \cap-G)^{0}=\emptyset$
by Problem 15 in Chapter 3, §12 and Problem 15 in Chapter 3, §16.]

Exercise $$\PageIndex{13}$$

("Simplified" uniform boundedness theorem.) Let $$f_{n}: (S, \rho) \rightarrow\left(T, \rho^{\prime}\right)$$ be continuous for $$n=1,2, \ldots,$$ with $$S$$ complete. If $$\left\{f_{n}(x)\right\}$$ is a bounded sequence in $$T$$ for each $$x \in S,$$ then $$\left\{f_{n}\right\}$$ is uniformly bounded on some open $$G \neq \emptyset:$$
$(\forall p \in T)(\exists k)(\forall n)(\forall x \in G) \quad \rho^{\prime}\left(p, f_{n}(x)\right) \leq k.$
[Outline: Fix $$p \in T$$ and $$(\forall n)$$ set
$F_{n}=\left\{x \in S |(\forall m) n \geq \rho^{\prime}\left(p, f_{m}(x)\right)\right\}.$
Use the continuity of $$f_{m}$$ and of $$\rho^{\prime}$$ to show that $$F_{n}$$ is closed in $$S,$$ and $$S=\bigcup_{n=1}^{\infty} F_{n}$$. By Theorem 1, $$S$$ is nonmeagre; so at least one $$F_{n}$$ is not nowhere dense-call it $$F$$, so $$(\overline{F})^{0}=F^{0} \neq \emptyset$$. Set $$G=F^{0}$$ and show that $$G$$ is as required.]

Exercise $$\PageIndex{14}$$

Let $$f_{n}: (S, \rho) \rightarrow\left(T, \rho^{\prime}\right)$$ be continuous for $$n=1,2, \ldots$$. Show that if $$f_{n} \rightarrow f$$ (pointwise) on $$S,$$ then $$f$$ is continuous on $$S-Q,$$ with $$Q$$ meagre in $$S.$$
[Outline: $$(\forall k, m)$$ let
$A_{k m}=\bigcup_{m=n}^{\infty}\left\{x \in S | \rho^{\prime}\left(f_{n}(x), f_{m}(x)\right)>\frac{1}{k}\right\}.$
By the continuity of $$\rho^{\prime}, f_{n}$$ and $$f_{m}, A_{k m}$$ is open in $$S.$$ (Why?) So by Problem 12, $$\bigcup_{m=1}^{\infty}\left(\overline{A_{k m}}-A_{k m}\right)$$ is meagre for $$k=1,2, \ldots$$.
Also, as $$f_{n} \rightarrow f$$ on $$S, \bigcap_{m=1}^{\infty} A_{k m}=\emptyset.$$ (Verify!) Thus
$(\forall k) \quad \bigcap_{m=1}^{\infty} \overline{A_{k m}} \subseteq \bigcup_{m=1}^{\infty}\left(\overline{A_{k m}}-A_{k m}\right).$
(Why?) Hence the set $$Q=\bigcup_{k=1}^{\infty} \bigcap_{m=1}^{\infty} \overline{A_{k m}}$$ is meagre in $$S.$$
Moreover, $$S-Q=\bigcap_{k=1}^{\infty} \cup_{m=1}^{\infty}\left(-A_{k m}\right)^{0}$$ by Problem 16 in Chapter 3, §16. Deduce that if $$p \in S-Q,$$ then
$(\forall \varepsilon>0)\left(\exists m_{0}\right)\left(\exists G_{p}\right)\left(\forall n, m \geq m_{0}\right)\left(\forall x \in G_{0}\right) \quad \rho^{\prime}\left(f_{m}(x), f_{n}(x)\right)<\varepsilon.$
Keeping $$m$$ fixed, let $$n \rightarrow \infty$$ to get
$(\forall \varepsilon>0)\left(\exists m_{0}\right)\left(\exists G_{p}\right)\left(\forall m \geq m_{0}\right)\left(\forall x \in G_{p}\right) \quad \rho^{\prime}\left(f_{m}(x), f(x)\right) \leq \varepsilon.$
Now modify the proof of Theorem 2 of Chapter 4, §12, to show that this implies the continuity of $$f$$ at each $$p \in S-Q$$.]