MAT3, Mat-Øk 3, Mat-Tek3 -- Efterår 2014

Analyse 1

Onsdag den 1. oktober 2014,  kl. 08:15
Sted: Rum NJV14, 4-117.

Dagens program

08:15-10:15

Repetition og forelæsning: kompakte mængder, Heine-Borel sætningen, kontinuitet i metriske rum.

10:30--12:00

Exercises: page 156 in [Lay], exercises 3, 5 and 8.
Hint for exercise 3:
a). Let $O_n=]0,3-1/n[$ for $n\geq 1$ and consider $\cup_{n\geq 1} O_n$.
b). Try $O_n=]0,2-1/n[ \cup ]3+1/n,5[$.
d). Try $O_n=]1/n, 4[$.
e). The set of rationals is countable, thus the set $S:=\{x\in \mathbb{Q}: 0\leq x\leq 2\}$ can be indexed as $S=\{q_1, q_2, q_3, ...\}$. Since $\sqrt{2}$ is irrational, we have $r_n=|q_n-\sqrt{2}|/2>0$ for all $n$. Define $O_n=]q_n-r_n,q_n+r_n[$. Prove that $S\subset \cup_{n\geq 1} O_n$ but no finite union can cover $S$ (one of the ingredients to be used is that there exists a rational number between any two real numbers).

Hint for exercise 5:
a). Let $S=S_1\cup S_2$. Assume that $O_\alpha$ with $\alpha\in F$ are open sets, and $S\subset \cup_{\alpha\in F}O_\alpha$. Since $S_1\subset S$ and $S_1$ is compact we can find a finite $F'\subset F$ such that $S_1\subset \cup_{\alpha\in F'}O_\alpha$. Also, since $S_2\subset S$ and $S_2$ is compact we can find a finite $F''\subset F$ such that $S_2\subset \cup_{\alpha\in F''}O_\alpha$. Then we have $S\subset \cup_{\alpha\in (F'\cup F'')}O_\alpha$ with $(F'\cup F'')\subset F$ finite, and we are done.
b). Remember that an unbounded set cannot be compact.

Hint for exercise 8:
a). Assume that $O_\alpha$ with $\alpha\in F$ are open sets, and $T\subset \cup_{\alpha\in F}O_\alpha$. Because $T\cup T^c$ is the whole space, we must have $S\subset \cup_{\alpha\in F}(O_\alpha \cup T^c)$. Since $T$ is closed, we have that $T^c$ is open and $A_\alpha:=O_\alpha \cup T^c$ is also open. Because $S\subset \cup_{\alpha\in F}A _\alpha$ and $S$ is compact, we can extract a finite $F'\subset F$ such that $S\subset \cup_{\alpha\in F'}A _\alpha =(\cup_{\alpha\in F'}O_\alpha) \cup T^c$. Since $T\subset S$ we must have $T \subset \cup_{\alpha\in F'}O_\alpha$ and we are done.

Literatur: afsnit 3.4 og kapitel 4 fra 'Noter for Analyse 1 og 2'.