A curve is a 1-dimensional space. Even though you draw a picture of it in a 2 (or maybe higher?) dimensional space, you should think of the set of points in the curve as being the only points that we care about.

In a closed curve, you can never fall out of the space by moving around in it. In a curve with endpoints, you can fall out of the space by walking across one of the endpoints, so the endpoints are considered to be boundary components.

A closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoints.

It's important to distinguish having a boundary vs. being the boundary of something, since in some sense the difference between the two is the whole point of homology.

For a one-dimensional shape, a boundary point is one you can stand on and move in one direction, but not in the other. So for an open line segment 0 < x < 1, there is no boundary, since at every point you can move in both directions. For a half-open line segment 0 < x <= 1 the point x=1 is the boundary, since you can move left but not right.

For a two-dimensional shape (or more formally "manifold"), an interior point is one which locally looks like the plane (you can move in all directions), and a boundary point is one which locally looks like a half-plane. All the boundary points taken together will be a one-dimensional manifold _without boundary_, which is pretty neat. For example, take an annulus (a 2D shape like a CD-rom). The boundary of the annulus is two one-dimensional manifolds (the boundary circles), and those circles have no boundary.

The boundary of a line segment is its end points. Intuitively, it a line doesn't have end points, then it's a loop i.e. a closed curve without boundary

I've definitely been deep into some topology proofs where I suddenly proved a true fact couldn't possibly be true and then I realized I'd forgotten that not open doesn't imply not closed or something. It's funny the tricks that plays on your brain even when you know better.

Think of a curve as a railroad track, not the area that it encloses. A closed curve has no ends.

The disk that it encloses, indeed, has a boundary (the curve in question).

That's what open and curve mean in topology. For example, the set of x with 0 <= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously confusing terminology, since sets can be both closed and open, or neither...

>That's what open and curve mean in topology. For example, the set of x with 0 <= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously confusing terminology, since sets can be both closed and open, or neither...

You have it backwards; [0, 1] is closed and (0, 1) is open. This gives the terminology "open interval" and "closed interval".

The confusion about "why does a closed curve have no boundary" likely comes more from the word "boundary". The point is that for an n-dimensional object, the boundary is (n-1)-dimensional. For a curve, which is 1-dimensional, the boundary is 0-dimensional, ie points-- so we're looking for endpoints, and a closed curve doesn't have any.

Hm? That looks backwards. [0,1] is a closed interval, (0,1) is an open interval. The former includes its boundary, the latter doesn’t.

But a closed curve doesn’t mean the same thing as a curve which is a closed set. A closed curve is a continuous image of S^1 (the circle) , yeah?

The author means something different. He uses "open curve" to mean a curve with endpoints, and "closed curve" to mean one without.