1. ## Real Number Analysis: About Metric Spaces

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 $E$ is a subset of a metric space $\left ( X,d \right )$.

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 $E'$ is closed.
3. Prove that the derived sets of $E$ and of the closure of $E$ coincide.
4. Do $E$ and $E'$ have the same set of limit points?
5. Prove that the closure of $E$ is a closed set.

6. Construct a bounded set of real numbers with exactly three limit points.
7. Let $X$ be an infinite set. For $p$ an element of $X$ and $q$ an element of $X$, define $d \left ( p,q \right ) = 1$ if $p \not= q$, and $d \left( p,q \right ) = 0$ if $\left (p = 1 \right )$. Prove that $\left ( X,d \right )$ 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 $E$ contained in $R^k$ convex if:

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

when $x \in E$, $y \in E$ and $0 < \lambda < 1$.
• $E'$ is the set of limit points of $E$.
• A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \not= p$ such that $q \in E$.
• A neighborhood of a point $p$ is a set $N_r \left ( p \right )$ consisting of all points $q$ such that $d \left ( p, q \right ) < r$. The number $r$ is called the radius of $N_r \left ( p \right )$.

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!

2. Can I get this thread moved to the Calculus section? I just noticed that the calculus section includes real analysis.