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 $p \in S-Q$.]