6.8.E: Problems on Baire Categories and Linear Maps
Verify the equivalence of the various formulations in Definition 1. Discuss: \(A\) is nowhere dense iff it is not dense in any open set \(\neq \emptyset\).
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).]
Complete the missing details in the proof of Theorems 1 to 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.
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.]
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.]
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!
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)).
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).]
(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 \(S\).
(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.]
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.]
("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.]
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 \(p \in S-Q\).]