Can a Complete Metric Space Have Empty Interior?

[pivoxa15]

Homework Statement

Can a complete metric space have empty interior?

Homework Equations

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.

The Attempt at a Solution

But if M has no Cauchy sequence to start with or anything else for that matter (i.e have empty interior than it can also be labeled as complete? Or is my understanding lacking some important information?

 

[Dick]

A single point constitutes a trivial metric space. It's complete, open, closed, compact and lots of other things, too!

 

[pivoxa15]

A single point also has empty interior.

What about a metric space with 2 points? It still has empty interior.

 

[Dick]

A single point also has empty interior.

What about a metric space with 2 points? It still has empty interior.

True. Guess I'd better think again.

 

[Dick]

We'd better be a little careful here. Interior and exterior only have nontrivial meaning if we are speaking of the metric space as a subset of another space. If we are speaking of a single point space {x} in isolation then the interior of {x} is {x}. If we are speaking for example of {0} as a subset of the reals, then it has empty interior.

 

[pivoxa15]

Interior of the whole metric space is always non empty.

So the subspace of a complete metric space is compelete so has non empty interior? Since we could have a sequence of points starting in the large metric space and obtaining a limit in this subspace. Where this limit point can be in the interior of the subspace. Hence non empty interior for this subspace?

 

[AKG]

Can a complete metric space have empty interior?

The interior of a metric space X is X itself. So a metric space has empty interior iff that metric space is itself empty. The empty set together with the empty function is a metric space.

 

[pivoxa15]

Is the empty set also complete?

 

[Dick]

Is the empty set also complete?

pivoxa15, this is a sad moment. Think!

 

[pivoxa15]

From the definition
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.

I'd say yes because there is no Cauchy sequence or any sequence in the empty set.

 

[Dick]

I would agree. But are these void case problems really that interesting? Is the empty set colorless?

 

[guitarra]

Interior of the whole metric space is always non empty.

So the subspace of a complete metric space is compelete so has non empty interior? Since we could have a sequence of points starting in the large metric space and obtaining a limit in this subspace. Where this limit point can be in the interior of the subspace. Hence non empty interior for this subspace?

That`s the right one