Editing Betti number (section)

== Geometric interpretation ==
[[File:Torus cycles.png|thumb|For a torus, the first Betti number is ''b''<sub>1</sub> = 2, which can be intuitively thought of as the number of circular "holes".]]
Informally, the ''k''th Betti number refers to the number of ''k''-dimensional ''holes'' on a topological surface. A "''k''-dimensional ''hole''" is a ''k''-dimensional cycle that is not a boundary of a (''k''+1)-dimensional object.

The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional [[simplicial complex]]es:
* ''b''<sub>0</sub> is the number of connected components;
* ''b''<sub>1</sub> is the number of one-dimensional or "circular" holes;
* ''b''<sub>2</sub> is the number of two-dimensional "voids" or "cavities".

Thus, for example, a torus has one connected surface component so ''b''<sub>0</sub> = 1, two "circular" holes (one equatorial and one [[Zonal and meridional|meridional]]) so ''b''<sub>1</sub> = 2, and a single cavity enclosed within the surface so ''b''<sub>2</sub> = 1.

Another interpretation of ''b''<sub>k</sub> is the maximum number of ''k''-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so ''b''<sub>1</sub> = 2.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211212/XxFGokyYo6g Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20200829013025/https://www.youtube.com/watch?v=XxFGokyYo6g&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web|last=Albin|first=Pierre|date=2019|title=History of algebraic topology|website=[[YouTube]]|url=https://www.youtube.com/watch?v=XxFGokyYo6g}}{{cbignore}}</ref>

The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions.