3.8.E: Problems on Neighborhoods, Open and Closed Sets (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
⇒1. Verify Example (1).
[ Hint: Given p∈Gq(r), let
δ=r−ρ(p,q)>0.(Why>0?)
Use the triangle law to show that
x∈Gp(δ)⇒ρ(x,q)<r⇒x∈Gq(r).]
⇒2. Check Example (2); see Figure 8.
[Hint: If ¯p∈(¯a,¯b), choose δ less than the 2n numbers
pk−ak and bk−pk,k=1,…,n;
then show that G¯p(δ)⊆(¯a,¯b).]
Prove that if ¯p∈G¯q(r) in En, then G¯q(r) contains a cube [¯c,¯d] with ¯c≠¯d and with center ¯p.
[Hint: By Example (1), there is G¯p(δ)⊆G¯q(r). Inscribe in G¯p(12δ) a cube of diagonal δ. Find its edge-length (δ/√n). Then use it to find the coordinates of the endpoints, ¯c and ¯d( given ¯p, the center). Prove that [¯c,¯d]⊆G¯p(δ).]
Verify Example (3).
[Hint: To show that no interior points of [¯a,¯b] are outside (¯a,¯b), let ¯p∉(¯a,¯b). Then at least one of the inequalities ak<pk or pk<bk fails. (Why?) Let it be a1<p1, say, so p1≤a1.
Now take any globe G¯p(δ) about ¯p and prove that it is not contained in [¯a,¯b] (so ¯p cannot be an interior point). For this purpose, as in Problem 3, show that G¯p(δ)⊇[¯c,¯d] with c1<p1≤a1. Deduce that ¯c∈G¯p(δ), but ¯c∉[¯a,¯b]; so G¯p(δ)⊈[¯a,¯b].]
Prove that each open globe G¯q(r) in En is a union of cubes (which can be made open, closed, half-open, etc., as desired). Also, show that each open interval (¯a,¯b)≠∅ in En is a union of open (or closed) globes.
[Hint for the first part: By Problem 3, each ¯p∈G¯q(r) is in a cube Cp⊆G¯q(r). Show that G¯q(r)=⋃Cp.]
Show that every globe in En contains rational points, i.e., those with rational coordinates only (we express it by saying that the set Rn of such points is dense in En); similarly for the set In of irrational points (those with irrational coordinates).
[Hint: First check it with globes replaced by cubes (¯c,¯d); see §7, Corollary 3. Then use Problem 3 above. ]
Prove that if ¯x∈G¯q(r) in En, there is a rational point ¯p (Problem 6) and a rational number δ>0 such that ¯x∈G¯p(δ)⊆G¯q(r). Deduce that each globe G¯q(r) in En is a union of rational globes (those with rational centers and radii). Similarly, show that G¯q(r) is a union of intervals with rational endpoints.
[Hint for the first part: Use Problem 6 and Example (1).]
Prove that if the points p1,…,pn in (S,ρ) are distinct, there is an ε>0 such that the globes G(pk;ε) are disjoint from each other, for k=1,2,…,n.
Do Problem 7, with G¯q(r) replaced by an arbitrary open set G≠∅ in En.
Show that every open set G≠∅ in En is infinite (∗ even uncountable; see Chapter 1,§9 ).
[Hint: Choose G¯q(r)⊆G. By Problem 3,G¯p(r)⊃L[¯c,¯d], a line segment.]
Give examples to show that an infinite intersection of open sets may not be open, and an infinite union of closed sets may not be closed.
[Hint: Show that
∞⋂n=1(−1n,1n)={0}
and
∞⋃n=2[1n,1−1n]=(0,1).]
Verify Example (6) as suggested in Figures 9 and 10.
[Hints: (i) For ¯Gq(r), take
δ=ρ(p,q)−r>0.(Why>0?)
(ii) If ¯p∉[¯a,¯b], at least one of the 2n inequalities ak≤pk or pk≤bk fails (why?), say, p1<a1. Take δ=a1−p1.
In both (i) and (ii) prove that A∩Gp(δ)=∅ (proceed as in Theorem 1).]
Prove the last parts of Theorems 3 and 4.
Prove that A0, the interior of A, is the union of all open globes contained in A (assume A0≠∅). Deduce that A0 is an open set, the largest contained in A.
For sets A,B⊆(S,ρ), prove that
(i) (A∩B)0=A0∩B0;
(ii) (A0)0=A0; and
(iii) if A⊆B then A0⊆B0.
[ Hint for (ii):A0 is open by Problem 14.]
Is A0∪B0=(A∪B)0?
[Hint: See Example (4). Take A=R,B=E1−R.]
Prove that if M and N are neighborhoods of p in (S,ρ), then
(a) p∈M∩N;
(b) M∩N is a neighborhood of p;
*(c) so is M0; and
(d) so also is each set P⊆S such that P⊇M or P⊇N.
[ Hint for (c): See Problem 14.]
The boundary of a set A⊆(S,ρ) is defined by
bdA=−[A0∪(−A)0];
thus it consists of points that fail to be interior in A or in −A.
Prove that the following statements are true:
(i) S=A0∪ bd A∪(−A)0, all disjoint.
(ii) bdS=∅, bd ∅=∅.
∗( iii )A is open iff A∩ bd A=∅;A is closed iff A⊇ bd A.
(iv)InEn,
bdG¯p(r)=bd¯G¯p(r)=S¯p(r)
(the sphere with center ¯p and radius r). Is this true in all metric spaces?
[Hint: Consider Gp(1) in a discrete space (S,ρ) with more than one point in S; see §11, Example (3).]
(\mathrm{v}) \operatorname{In} E^{n}, if (\overline{a}, \overline{b}) \neq \emptyset, then
\operatorname{bd}(\overline{a}, \overline{b}]=\operatorname{bd}[\overline{a}, \overline{b})=\operatorname{bd}(\overline{a}, \overline{b})=\operatorname{bd}[\overline{a}, \overline{b}]=[\overline{a}, \overline{b}]-(\overline{a}, \overline{b}).
(vi) \operatorname{In} E^{n},\left(R^{n}\right)^{0}=\emptyset ; hence bd R^{n}=E^{n}\left(R^{n} \text { as in Problem } 6\right).
Verify Example ( 8 ) for intervals in E^{n}.