system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Why do cysts keep coming back over and over if the sac isn't removed? Does the sac generate more sebacum (sp?) or is it something else?
gpt
To answer your question, it is important to define the terms which describe what laypeople call "cysts."

There are abscesses which are pus-filled pockets within the skin and associated soft tissue.  They form when bacteria get into the tissue and multiply.  Pus is a thick, typically off-white fluid which is made of dead bacteria, body fluids, and dead white blood cells that are trying to fight the infection.

A sebaceous cyst (sometimes called an epidermoid cyst) is a sterile (not infected) pocket of material under the skin.  It is like an abscess but not filled with pus.  There are several different types but the difference is not important for this explanation.  They form most commonly when skin cells start growing underneath the layer they are supposed to.  They form a round structure that fills up with dead skin cells.  Usually when skin cells die, they flake off the body but since these skin cells are trapped under the skin, the dead cells fill up the cyst.  The material inside a sebaceous cyst is called sebum and it is much thicker than pus; sebum is like toothpaste while pus is like gravy.

You treat abscesses and sebaceous cysts differentially.  An abscess must be cut into so pus can drain.  This takes pressure off the surrounding tissue, improves blood flow, and allows white blood cells, healing cells, and antibiotics into the area.  An abscess will not heal until it drains because the bacteria can hide inside the pocket from white blood cells and antibiotics.

On the other hand, if you try to remove the sebum from a sebaceous cyst (I say remove because it is so thick that it will not drain on its own even if you cut the cyst open), it will decrease the size of the cyst temporarily but it can recur.  This is because the skin cells that formed the sebum in the first place are still there.  They are the "sac" that you referred to in your question.

As an aside, sometimes sebaceous cysts can become inflamed if bacteria gets inside the sebum.  In those cases, doctors typically prescribe antibiotics and allow the infection to resolve before attempting to cut the sac out.  It is difficult to remove an inflamed sebaceous cyst because the sac becomes very fragile and will tear into multiple little pieces rather than come out in one or a few large pieces.  If you leave just a single tiny portion behind, then the entire cyst can return.

For bonus material, people can develop recurrent abscesses in the same place as well.  This is not due to failure to fully remove a "sac" (there is none in an abscess), but rather due to scar tissue that forms in the skin and underlying tissue.  If that area becomes infected again, an abscess has an easier time forming in the same location because of that scar tissue, like grooves that form in the couch cushion encourage you to sit in that spot again.