Analysis Revisited: 6 Questions & Solutions to Prove Metric Space Properties

[Krizalid1]

I'll be posting Analysis questions. :D
If you submit a solution, please use Spoiler. If you have an alternate one, use the Spoiler anyway.

If this goes right, I'll be making a big PDF containing all solutions so that it may be useful for students that are taking this course.

1. Let $(X,d)$ be a metric space and $0<\alpha\le1.$ Let $\phi$ be defined in $X\times X$ by $\phi(x,y)=d(x,y)^\alpha.$ Prove that $\phi$ is a metric. What happens if $\alpha>1$?

2. Let $(A_n)_n$ be a decreasing sequence of non-empty bounded sets of a metric space. Is it true that $A_1\cap\cdots\cap A_n\cap\cdots\ne\varnothing$?

3. Show that all set of positive real numbers that contains the zero is the image of a distance function.

4. Let $f:[0,\infty)\to[0,\infty)$ be a function such that $f\circ d$ is a metric, for all $d$ metric. Show that if $a\ge0,$ $f(a)=0$ iff $a=0.$ Also show for all $a,b,c\in[0,\infty)$ such that $|a-b|\le c\le a+b$ we have $f(a)\le f(b)+f(c).$

5. Show an example of a metric space which exists a closed ball whose diameter is less that its radius.

6. Let $(X,d)$ be a metric space and $A,B\subseteq X.$

1. Show that if $A\subseteq B$ then $\text{diam}(A)\le\text{diam}(B).$
2. Show that $\text{diam}(A\cup B)\le\text{diam}(A)+\text{diam}(B)+d(a,b)$ for any $a\in A$ and $b\in B.$
3. Show that $X$ is bounded iff all countable subset of $X$ is bounded.

4. Let $(X,d)$ be a metric space. A subset $A$ of $X$ is called totally bounded if for each $\epsilon>0$ exist $x_i\in X_i,$ $i=1,\ldots,n$ such that $A\subseteq B_\epsilon(x_1)\cup\cdots\cup B_\epsilon(x_n).$

1. Show that all finite set is totally bounded.
2. Show that all totally bounded set is bounded and that exist bounded sets that aren't totally bounded ones.
3. Show that if $A\subseteq X$ is totally bounded then for each $\epsilon>0$ exist $a_i\in A,$ $i=1,\ldots,n$ such that $A\subseteq B_\epsilon(a_1)\cup\cdots\cup B_\epsilon(a_n).$
4. Show that a subset $A$ of $X$ is totally bounded iff for each $\epsilon>0$ exist a finite family $(A_i)_{i=1,\ldots,n}$ of subsets of $X$ with $\text{diam}(A_i)<\epsilon$ such that $A\subseteq A_1\cup\cdots\cup A_n.$

 

[Nono713]

Thanks Krizalid! This will be most helpful as revision, I hope you'll post some questions on uniform continuity and complex analysis as well :D however I find something unclear: for problem 3, do you mean we need to show that any set of positive real numbers including zero is the image of some distance function in some metric space? The "all" threw me off.

Problem 2:
[sp]This is only true if the nested sets are closed. If they are open, a counterexample in $\mathbb{R}$ is $A_n = (0, \frac{1}{n})$ - clearly this set is nonempty and bounded for all $n > 0$, yet the intersection of them all is empty, because for any $a > 0$ there exists an $n \in \mathbb{N}$ such that $a \not \in A_n$. If all the sets are closed, then it is true. The proof is as follows: there is at least one element $a_n \in A_n$, and since $A_n$ is contained in $A_1$, this gives us a sequence of elements $(a_n)_{n = 1}^\infty$ in $A_n$. Because $A_1$ is bounded, the sequence must be bounded too, so it has a subsequence converging to an element $a$. Because the sets are decreasing, this subsequence must eventually be in $A_n$ for any $n \in \mathbb{N}$, and because $A_n$ is closed, $a \in A_n$. Therefore $a \in \cap_{n = 1}^\infty A_n$.[/sp]

Problem 5:
[sp]The discrete metric defined as $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ otherwise on any set is such a metric space. The closed ball of radius $0.5$ centered on $x$ contains only $x$, and so it has diameter zero.[/sp]

I might do more later, problem 1 is tricky.