414/514 – Metric spaces | A kind of library

414/514 – Metric spaces

This is homework 2, due Monday September 26 at the beginning of lecture.

Let $(X,d)$ be a metric space.

Show that if $d_1:X\times X\to{\mathbb R}$ is defined by $\displaystyle d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$ , then $d_1$ is also a metric on $X$ .
Show that if $U$ is open in $(X,d)$ then it is open in $(X,d_1)$ , and viceversa.

Recall that $U$ is open iff it is a union of open balls. Use this to explain why it suffices to show that if $U$ is open in $(X,d)$ then for any $x\in U$ there is an $\epsilon>0$ such that

$B_\epsilon^{d_1}(x):=\{y\mid d_1(x,y)<\epsilon\}\subseteq U$ ,

and similarly, if $V$ is open in $(X,d_1)$ then for any $z\in V$ there is a $\delta>0$ such that

$B_\delta^d(z):=\{w\mid d(z,w)<\delta\}\subseteq V$ .

In turn, explain why to show this it suffices to prove that for any $x\in X$ and any $\eta>0$ there is a $\rho>0$ such that

$B^{d_1}_\eta(x)\supseteq B^d_\rho(x)$

and, similarly, for any $\tau>0$ there is a $\mu>0$ such that

$B_\tau^d(x)\supseteq B^{d_1}_\mu(x)$ .

Finally, prove this by showing that we can take $\rho=\epsilon$ (no matter what $x$ is) and similarly, find an appropriate $\mu$ that works for $\tau$ (again, independently of $x$ ).

Illustrate the above in ${\mathbb R}^2$ as accurately as possible.
Suppose that a sequence $(x_n)_{n\in{\mathbb N}}$ converges to $x$ in $(X,d)$ and to $x'$ in $(X,d_1)$ . Show that $x=x'$ .
Is it true that a sequence $(x_n)_{n\in{\mathbb N}}$ is Cauchy in $(X,d)$ iff it is Cauchy in $(X,d_1)$ ? (Give a proof or else exhibit a counterexample, with a proof that it is indeed a counterexample.)
$(*)$ Show that any dense $G_\delta$ subset of ${\mathbb R}$ has the same size as ${\mathbb R}$ .

This entry was posted on Monday, September 19th, 2011 at 9:34 am and is filed under 414/514: Analysis I. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

	ปั่นสล็อต on 414/514 The theorems of Rieman…
	Richard Kimberly Hec… on Woodin’s proof of the se…
	Is there a Borel-mea… on 414/514 Examples of Baire clas…
	Simple bijectiveness… on 580 -Some choiceless results…
	Schur’s theore… on 580 -III. Partition calcu…
	Uncountable linearly… on 403/503 – Homework …
	What is a non-constr… on On strong measure zero se…
	function from Nmathb… on Hindsight
	Cauchy Schwarz Inequ… on 275 -Positive polynomials
	What is a Cantor-sty… on 580 -Some choiceless results…
	Question about Gener… on 580 -Some choiceless results…
	Set has Measure Zero… on On strong measure zero se…
	Non-measurable subse… on 580 -Cardinal arithmetic …
	hmeng1 on 414/514 The theorems of Rieman…
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …

M	T	W	T	F	S	S
			1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30

A kind of library