Real Analysis -Open/Closed sets of Metric Spaces

[rad0786]

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...

Homework Statement

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?

Homework 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

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.

 

[StatusX]

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

 

[rad0786]

oh darn...

 

[rad0786]

Can we focus on part b) only.

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

Then it is open and closed?

 

[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).

 

[rad0786]

Oh okay...

I find the discrete metric very unusual

 

[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.