Betti Numbers: Understanding the Topology of Spaces

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In summary, betti numbers are a way to measure the topological properties of a space. The zeroth betti number represents the number of connected components in the space, and is always at least one because every space has at least one connected component. The first betti number represents the number of loops in the space, but this interpretation can be misleading as certain spaces like the projective plane and Klein bottle have different numbers of loops. The empty space has no connected components, but it is still considered connected.
  • #1
realanony87
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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 ?
 
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  • #2
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
yenchin said:
One can always call the collection of two distinct spheres as one space.
And two is at least one. :tongue:
 
  • #4
Can you give me a case where the zeroth Betti number is 0 and explain why please?
 
  • #5
Hurkyl said:
And two is at least one. :tongue:

Oops. I misunderstood the statement. :rofl: My bad.
 
  • #6
The empty space has no connected components. :)
 
  • #7
adriank said:
The empty space has no connected components. :)

the empty set is connected
 
  • #8
realanony87 said:
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?
 
  • #9
wofsy said:
the empty set is connected

Yes, it is connected, but it has no connected components. A connected component must be nonempty.
 
  • #10
adriank said:
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
 
  • #11
adriank said:
Yes, it is connected, but it has no connected components. A connected component must be nonempty.

Totally agree :)
 

1. What are Betti numbers?

Betti numbers are topological invariants that measure the number of independent cycles in a topological space. They are used to understand the shape and connectivity of a space.

2. How are Betti numbers calculated?

Betti numbers are calculated using homology groups, which are algebraic structures that capture the properties of a topological space. The number of Betti numbers is equal to the number of homology groups in the space.

3. What do Betti numbers tell us about a space?

Betti numbers provide information about the number of holes or voids in a space. They can also indicate the dimensionality of a space, as well as its connectivity and complexity.

4. How are Betti numbers used in scientific research?

Betti numbers have applications in a variety of fields, including physics, biology, and computer science. They are used to analyze and classify data, as well as to understand the behavior of complex systems.

5. Are Betti numbers applicable to all types of spaces?

Yes, Betti numbers can be calculated for any topological space, including manifolds, graphs, and networks. They are also useful for studying higher-dimensional spaces, such as hypersurfaces and simplicial complexes.

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