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

is a subset of a metric space

.

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

is closed.

3. Prove that the derived sets of

and of the closure of

coincide.

4. Do

and

have the same set of limit points?

5. Prove that the closure of

is a closed set.

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

7. Let

be an infinite set. For

an element of

and

an element of

, define

if

, and

if

. Prove that

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 contained in convex if:

- is the set of limit points of .

- A point is a limit point of the set if every neighborhood of contains a point such that .

- A neighborhood of a point is a set consisting of all points such that . The number is called the radius of .

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!