I never imagined that there would be a day where I would need to ask for help on mathematics, but sadly (for me anyway), today happens to be the day.

Enough, I'll cut to the chase. I have no idea how to do these problems:

Homework #3

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All problems assume that the set $\displaystyle E$ is a subset of a metric space $\displaystyle \left ( X,d \right )$.

The following problems only require what was covered through Thursday's lecture:

1. Show that each neighborhood in a metric space is convex.

(check text for definition).

2. Prove that $\displaystyle E'$ is closed.

3. Prove that the derived sets of $\displaystyle E$ and of the closure of $\displaystyle E$ coincide.

4. Do $\displaystyle E$ and $\displaystyle E'$ have the same set of limit points?

5. Prove that the closure of $\displaystyle E$ is a closed set.

6. Construct a bounded set of real numbers with exactly three limit points.

7. Let $\displaystyle X$ be an infinite set. For $\displaystyle p$ an element of $\displaystyle X$ and $\displaystyle q$ an element of $\displaystyle X$, define $\displaystyle d \left ( p,q \right ) = 1$ if $\displaystyle p \not= q$, and $\displaystyle d \left( p,q \right ) = 0$ if $\displaystyle \left (p = 1 \right )$. Prove that $\displaystyle \left ( X,d \right )$ is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

And definitions:

- Convex: We call a set $\displaystyle E$ contained in $\displaystyle R^k$ convex if:

$\displaystyle \lambda x + \left (1 - \lambda \right ) y \in E$

when $\displaystyle x \in E$, $\displaystyle y \in E$ and $\displaystyle 0 < \lambda < 1$.

- $\displaystyle E'$ is the set of limit points of $\displaystyle E$.

- A point $\displaystyle p$ is a limit point of the set $\displaystyle E$ if every neighborhood of $\displaystyle p$ contains a point $\displaystyle q \not= p$ such that $\displaystyle q \in E$.

- A neighborhood of a point $\displaystyle p$ is a set $\displaystyle N_r \left ( p \right )$ consisting of all points $\displaystyle q$ such that $\displaystyle d \left ( p, q \right ) < r$. The number $\displaystyle r$ is called the radius of $\displaystyle N_r \left ( p \right )$.

I blame the text (*Principles of Mathematical Analysis*, Rudin 1976) and the professor for the opacity of lessons.

If anyone can walk me through, or provide a quick guide, I'd be very, very grateful!