TITLE: How to tell what dimension an object is?
QUESTION [35 upvotes]: I was reading about dimensions and in the Wikipedia article it states the following:

In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but a single polar coordinate (the angle) would be sufficient, so the circle is $1$-dimensional even though it exists in the $2$-dimensional plane. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages."
Taken from Dimension article at Wikipedia.

How could I know if a square is a $2$D object (which I'm sure it probably is), or that a circle is $1$D (which previously I believed it to be $2$D)?
Is there a way for any object to tell what dimensional object it is?
I don't really know how to tag this question, sorry.

REPLY [13 votes]: While goblin's answer is right, its a little specific. A more common notion is covering dimension, which is both applies in more situations, and is a little easier to understand.
First we start with the notion of open set. An open set is one that does not contain any points in its boundary. For example, the set of points less than $1$ away from $(0,0)$ is open, because the boundary is the set of points exactly $1$ away from $(0,0)$, and that set is disjoint from the first one. The set of points less than or equal to $1$ away from $(0,0)$ is not open, since it contains some points, all of the points actually, that are in its boundary.
Now, if you take a line and cover it with blobs that are open sets, there will be some points covered by $2$ blobs. Since open sets don't contain their boundaries, if you try putting them right next to each other, they either have to overlap or have space in between them.  If you do the same with the plane, some points will be covered by $3$ blobs. If you the same with space, some will be covered by $4$, etc...
So basically, the dimension of space is the max number of blobs to cover a point minus $1$.
We need to take one more thing into account though. Take a circle. If you cover it with tiny blobs, some of its points will have to be covered by $2$ blobs, so it is $1$ dimensional. But also notice that you could cover the entire circle with one big blob, suggesting it is $0$ dimensional. For this reason, we require the blobs to be arbitrarily small.
But what if we don't have a notion of size in our space? That's fine. What we say is that a space is dimension $d$, if when you cover the entire thing with blobs that are open sets, you can shrink (replace with a subset that is also an open set) the open sets such that each point is covered by at most $d+1$ blobs.
Notice how all this required was a notion of boundaries of sets. We didn't need to define distance or anything else, just boundaries.
Also note that if we viewed the circle as a space in and of itself, this still works. The boundaries of a set of points on the circle is just the set of any endpoints it has. The circle's dimension will be the same whether considered as part of the plane or as a space unto itself.
See https://en.wikipedia.org/wiki/Lebesgue_covering_dimension for more details.