# Betti number

1. Jan 31, 2009

### realanony87

This is probably a silly question but I am not from a maths background. I am a bit confused about betti numbers. From what i know :
The zeroth betti number is the number of connected components in space
The first betti number is the number of loops in space
the second is the number of cavities in the space

However does this mean that the zeroth betti number is always atleast one since there is always one connected component ?

2. Jan 31, 2009

### yenchin

Why would there be always at least one connected component ? A topological space can have more than one components. One can always call the collection of two distinct spheres as one space.

3. Jan 31, 2009

### Hurkyl

Staff Emeritus
And two is at least one. :tongue:

4. Jan 31, 2009

### realanony87

Can you give me a case where the zeroth Betti number is 0 and explain why please?

5. Jan 31, 2009

### yenchin

Oops. I misunderstood the statement. :rofl: My bad.

6. Feb 1, 2009

The empty space has no connected components. :)

7. Feb 4, 2009

### wofsy

the empty set is connected

8. Feb 4, 2009

### wofsy

There is always one connected component.

But your interpretation of the first Betti numer as the number of loops is hasty. What is the first Betti number of the projective plane? What about the Klein Bottle? How many loops do they have?

9. Feb 4, 2009

Yes, it is connected, but it has no connected components. A connected component must be nonempty.

10. Feb 4, 2009

### wofsy

true, i just thought i'd clarify the terminology

11. Feb 18, 2009

### Take_it_Easy

Totally agree :)