# What is the difference between hole and cuff in topology?

I am watching the video about topology on YouTube. I confuse the notions of hole and cuff mentioned in the video.

In my understanding, hole penetrates the sphere while cuff just drills a portion of the sphere. In the example of the video, sweater has four holes. I don't know this means a hole or a cuff.

I think about other examples. A torus has one hole. A bowl has one cuff (I'm not sure). What about a straw, is it a torus or a sphere with two cuffs.

Could someone explain and give some examples? Thanks.

• – Emilio Novati May 31 '17 at 15:58
• The way I saw it, a "cuff" is what I would personally call a hole in the surface. You've actually cut a piece of skin away from a surface. The "hole" like a donut hole is not a hole in the n-dimensional surface but a property of the surface, visible in the n+1 dimension but not in the n-dimension unless you take paths or cross-section cuts. – fleablood May 31 '17 at 16:13

My understanding:

For simplicity we are viewing finite 2-dimensional surfaces "floating" in 3-dimensional space.

Some, such as a closed disc or an multi-holed anulus, have boundaries and some, such as a sphere or multi-holed tori, do not. A torus has a "donut hole" which is intuitively obvious to see but very difficult to define. The thing is, a hole is can not be viewed locally "from" the surface itself, but must be viewed as how the surface itself is "shaped" in space.

A "hole" is a 3-dimensional hole in the space the surface occupies. If you think of the 2-dimensional surface as the "skin" of a 3-dimensional solid, the solid has a hole in it but the skin is intact.

A "cuff" is a 2-dimensional hole that is in the surface itself. If you cut a hole out of the skin of a sphere and the insides can now leak out.

The subtle part is that a cuff need not appear as a hole in the skin, it could be the border of a surface. Imagine a sphere. Cut a little patch of it and you have a giant bubble with a hole in it. But stretch the hole out and you have a bowl with a rim. Flatten it out and you simply have a closed disc. The cuff has become a border.

Consider a sphere. Cut two cuffs in it. Stretch those cuffs apart so that that remaining surface becomes cylinderical. Then you have a straw. Shrink one cuff and widen the other and flatten and you have an anulus.

Interestingly if you glue the two cuffs of a straw together you get a torus.

• I think I have understood. Thank you! – Jun May 31 '17 at 18:16
• Again. That is only my understanding. I'm not too familiar with the term "cuff" and I always took "hole" to be informal and imprecise. – fleablood May 31 '17 at 20:06