Lecture Notes

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====== Xingjian Tao's Notes and Question List ======
===== Notes =====
I made notes of some Chapters on LaTex. You can see the PDF here {{ math104-s21:s:math104_notes.pdf |}}. The organization of definitions and theorems is mainly based on Rudin, though supplemented by Ross. Sometimes I tried to rewrite the contents in a more symbolic way, so there maybe minor mistakes.


===== Question List =====
1. How is the definition of limit points related with the concept of limit?

Answer: An element is a limit point of $E$ iff. it is the limit of some inconstant sequence of points in $E$. Inconstant is important because the definition of limit points includes hollow neighborhoods.

2. Suppose $E$ is an infinite subset of a set $K$. Then $E$ has a limit point in $K$ iff. $K$ is compact. Prove this.

Answer: $\Longrightarrow$ can be found in 2.37 Theorem of Rudin. $\Longleftarrow$ can be found in Excercise 26 of Rudin Chapter 2.

3. If $f$ is a 1-1 continuous function on some interval $I$, prove $f^{-1}$ is continuous.

Answer: 1-1 continuous on an interval $\Longrightarrow$ $f$ is strictly increasing or decreasing and continuous $\Longrightarrow$ $f^{-1}$ is continuous.

4. If $f$ is a 1-1 continuous function on an open interval $I$, prove that $f(I)$ is open.

Answer:  According to Question 3, $f^{-1}$ is continuous. We know that $f^{-1}$ is continuous iff. for every open sets $O\subset I$, $f(O)$ is open. Since $I$ is open, we have $f(I)$ is open. (I think there maybe some mistakes so that I may only get $f(I)$ is open in $f(I)$ which is useless through my approach. But this proposition cannot be wrong since it's a statement in the proof of Ross 29.9 Theorem)\\
Oh I See. We can first prove $f$ must be strictly montonic function and then $f(I)$ is open.
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