A *boundary* is that which is an extremity of anything.

A *figure* is that which is contained by any boundary or boundaries.

The modern subject of topology studies space in a different way than geometry does. The geometric concepts of straightness, distance, and angle are excluded from topology, but the concept of boundary is central to topology. In topology, a sphere remains a sphere even when it’s squeezed or stretched.

Not everything has a boundary. For instance, the circumference of a circle has no boundary. Also a spherical surface has no boundary. In topology, a finite region with no boundary is called a *cycle.* Circles and spherical surfaces are cycles. In general, if something is a boundary, it has no boundary itself. So boundaries are cycles. But not all cycles are boundaries.

Topology uses cycles and boundaries to distinguish various spaces. For instance, on a spherical surface, every circle is the boundary of a region on that surface. But on a toroidal surface (rotate a circle around a line in the plane of the circle that doesn’t meet the circle), there are circles (for instance, that circle mentioned parenthetically) that don’t bound any region on the surface. Thus, spherical surfaces are topologically different from toroidal surfaces.

Other figures may be considered if other ambient spaces are allowed, although Euclid only uses plane and solid figures. For a one-dimensional example, a line segment with its endpoints as its boundary could be considered to be a figure in an infinite line. Also, a hemisphere could be considered to be a figure on the surface of a sphere with the equator as its boundary.

Nowhere in the *Elements* does a nonconnected figure occur. It’s apparent that figures are supposed to be connected.

For an example of a nonconnected figure, consider the following. Given a circle and a line that doesn’t intersect that plane, when that circle is rotated around the line in space, a solid results called a torus. The intersection of that torus with the original plane is the figure that consists of the original circle and another on the other side of the line. Considered as a single figure it is disconnected. It would be called two figures in the *Elements*.

Nonsimply-connected figures are those with holes in them. An example of such a figure is an annulus, also called a ring, which is the figure between two concentric circles. See proposition XII.16 for an illustration. There’s no indication that Euclid considered nonsimply-connected figures.

Nowhere in the *Elements* does a nonconvex figure explicitly occur. A pentagram is a five-pointed star. See the Guide to proposition IV.11 for an illustration. To study nonconvex polygons, Euclid would have had to admit angles greater than 180°, which he didn’t.