# Gaussian surface and closed surfaces

What is (intuitively speaking) a closed surface?

(this question may seem trivial but I think it's not so clear w.r.t. topology. Some definitions on Wikipedia seem confusing. It is said that a sphere is a closed surface since it has "no boundary" (<-what does that mean? for me it "intuitively" has a boundary) but the cylinder would not be a closed surface? I cannot understand that)

-> confusing statement from https://en.wikipedia.org/wiki/Surface_(topology)

A closed surface is a surface that is compact and without boundary. Examples are spaces like the sphere, the torus and the Klein bottle. Examples of non-closed surfaces are: an open disk, which is a sphere with a puncture; a cylinder, which is a sphere with two punctures;

So the requirement for a surface to be closed is that it (1) is compact and (2) has no boundary. Compactness is a topological property: a surface or other topological set is said to be compact if "any open cover of the set admits a finite subcover", and a boundary is defined as "the complement of the interior of a set within that set itself". Admittedly, these definitions might be somewhat abstract if you are not very familiar with topology.

Intuitively, a boundary of a surface can be seen as the set of points that have "points inside the surface" on one side and "points outside the surface" on the other side. An example of a surface with boundary is the closed disk, defined as the set of points in $\mathbb{R}^2$ with distance to the origin smaller than or equal to some parameter $d$. The boundary of this surface is the outer circle of the disk lying exactly at a distance $d$ from the origin. At one side of the circle are points within the disk; on the other side, there are points not lying inside the disk. You could imagine a two-dimensional ant "walking off the disk" from the boundary.

On the other hand, a sphere, when viewed purely as a two-dimensional surface, indeed does not have such a boundary, because there are no points from which the ant might leave the sphere, as long as it keeps walking two-dimensionally.

That the open disk and the cylinder are not closed surfaces is not because they have a boundary (both do not have one), but because they are not compact. The notion of compactness is somewhat harder to get a precise intuition for; to many, the open disk, which is not compact in the strict mathematical sense of the word, looks very compact.

Very heuristically speaking, a surface is compact if it (1) is not infinitely big and (2) it has no "soft edges". The cylinder fails to be compact because it is infinitely big. (Finitely big versions of the cylinder might be compact, depending on what the edges look like.) The open disk can be defined as the set of points in $\mathbb{R}^2$ at a distance from the origin strictly smaller than some $d$;. It is not compact because it does have "soft edges": our brave ant, while walking on the disk, would not be able to find the precise boundary of the disk (there is none), but it would still be able to walk off it, at least when the disk is embedded into $\mathbb{R}^2$.

In simple terms, imagine a region of space $A$ inside a closed surface.
Outside the region $A$ inside the closed surface is another region $B$.
If the surface is closed to move from region $A$ to region $B$ you have to cross the closed surface (boundary).
So if the closed surface is the surface of a sphere to get from inside the sphere to outside the sphere you must go through the surface.
A hollow cylinder with no ends is not a closed surface but a hollow cylinder with ends is a closed surface.

Here is a possible way to think about a closed surface or to test if a surface is closed.

By definition, in simple terms, a closed surface should contain some volume. This volume could be filled with some material (rock) or it could be, roughly speaking, empty (ping pong ball).

So, here is the test that should hold for a closed surface: if you are inside the volume contained by a closed surface, you cannot get out and, if you are outside that volume, you cannot get in - without breaking through the surface.

For instance, if you take a sealed cylindrical tin can, you cannot get anything out or insert anything in without making a hole in its surface or opening it. So, this is an example of a volume contained by a closed surface.

If you take a piece of a (cylindrical) PVC pipe, you can put things inside the pipe and take them out without breaking through its surface. So, we can say that the outer surface of the PVC pipe is not a closed surface, because it does not seal the volume inside the pipe.

On the other hand, if we look at the volume inside the walls of a PVC pipe, we can say that this volume is fully contained by total surface of the pipe, which includes the outer surface, the inner surface and the surface of the edges at the two ends of the pipe. You cannot insert anything into the walls of the pipe without breaking that surface somewhere. So, from this perspective, the surface of this cylindrical pipe is closed.

You can try to use this approach with other objects and see if it helps differentiate closed surface from non-closed surfaces.