>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.