Homework Help: Real Analysis -Open/Closed sets of Metric Spaces

1. Dec 10, 2006

So I have an exam in Real Analysis I coming up next week and I was hoping if someone can help me out.

I hope my question makes sense because I think I might be confused with defining the metric space or so...

1. The problem statement, all variables and given/known data

a)Suppose that we have a metric space M with the discrete metric

d(x,y) = 1 if x = y
d(x,y) = 0 if x =/= y

Is this open or closed?

b)Suppose that we are in R (the real line) and the metric is define as

d(x,y) = 1 if x = y
d(x,y) = 0 if x =/= y

Is this open or closed?

2. Relevant equations

Definition:

A set is Y open if every point in Y is an interior point
A set is Y closed if every point in Y is an limit point

3. The attempt at a solution

a)Im not even sure if question a makes sense because I didn't define the metric.

b) I'm pretty sure it is open and closed because both the definitions work.

2. Dec 10, 2006

StatusX

Is what open/closed? You haven't identified a set, only the metric. Also I think you have the definiton of the discrete metric backwards.

3. Dec 10, 2006

oh darn...

4. Dec 10, 2006

Can we focus on part b) only.

What if the set was just R (the entire real line)

Then it is open and closed?

5. Dec 10, 2006

StatusX

Remember, a set is only open or closed relative to a given topology. For a metric space, there is a natural induced topology from the metric. But for your last question the topology doesn't matter: for any topology on a space X, X is, by definition, open (and closed) in the topology.

In fact, the discrete metric induces the discrete topology, in which every subset is open (and closed).

6. Dec 10, 2006

Oh okay...

I find the discrete metric very unusual

7. Dec 10, 2006

StatusX

It's not that complicated. Every point has a ball containing it and no other points (eg, of radius 1/2), which just means points are open sets. Since unions of open sets are open, this means all sets are open.

The picture I get is sort of a lattice of isolated points. Also, don't get too hung up on the distances actually all being 1. This may be hard to visualize (if there are more than 4 points it's impossible to embed them in 3D space so they're all a distance 1 from every other point). But all that matters for most purposes is that it induces the discrete topology.