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 \(p \in S-Q\).]