2.8 Topological Transformations

Topological invariants, M bius strip, Klein bottle, sidedness, simple arcs, knottedness, nonlinear transformations

We now make a brief but significant departure from the world of linear transformations to that of topological transformations, which in all their generality are distinctly nonlinear. The properties of figures that are invariant under transformations that are continuous both ways form the study of topology. What does this mean? First, the ground rules: A topological transformation may stretch, bend, twist and compress a figure, but may not tear it, puncture it, nor cause it to be self-intersecting. So, "cubical" and "spherical" are not topological properties, because we can continuously deform either of these surfaces into the other. However, the "knottedness" of a closed loop of string incorporating a square knot is a topological property, because we cannot undo the knot using only the allowable topological deformations.

What other kinds of properties are possibly left invariant under such a deformational onslaught? Surprisingly, there are several, and they are of great interest and importance to geometers. For example, topological transformations preserve linear order and cyclic order. Lines, parabolas and the branches of a hyperbola belong to a class of topologically equivalent figures we call simple arcs. The property of being a closed or open curve or surface is a topological invariant. One-sidedness (a M bius strip or Klein bottle, for example) or two-sidedness are topologically invariant properties of surfaces. However, topology is a geometry in which size and shape have no meaning.

If a topological transformation is...