Betti Numbers: Understanding the Topology of Spaces

[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 ?

 

[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.

 

[Hurkyl]

One can always call the collection of two distinct spheres as one space.

And two is at least one.

 

[realanony87]

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

 

[yenchin]

And two is at least one.

Oops. I misunderstood the statement. My bad.

 

[adriank]

The empty space has no connected components. :)

 

[wofsy]

The empty space has no connected components. :)

the empty set is connected

 

[wofsy]

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 ?

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?

 

[adriank]

the empty set is connected

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

 

[wofsy]

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

true, i just thought i'd clarify the terminology

 

[Take_it_Easy]

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

Totally agree :)