BUT WE’LL TRY TO EXPLAIN TOPOLOGY DEFINITIONS: 1. The study of the properties of geometric figures or solids that are not normally affected by changes in size or shape. 2. The branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence ITS OKAY, NEITHER DO WE! BUT WE’LL TRY TO EXPLAIN ???DON’T UNDERSTAND???

EXPLAINING THE UNEXPLAINABLE STANDARD GEOMETRY VS PROJECTIVE GEOMETRY “While geometry is concerned with properties like absolute position, distance, and parallel lines, topology is concerned with properties like relative position and general shape.” Topology is the branch of mathematical study concerned with the nature and properties of geometrical figures. Topology was a term invented by Solomon Lefschetz, a mathematician who was connected with early topology. Topology has frequently been called rubber-band, rubber-sheet, or rubber-space geometry; it deals with the properties of geometric figures in space that remain unchanged when the space is changed (i.e. bent, twisted, or stretched); the only exceptions are that tearing the space is not allowed, and distinct points in the space cannot be made to coincide. Straightness and linear and angular measure of the plane are some of the properties that are obviously not maintained if the plane is distorted.

EARLY TOPOLOGY An example of an early topological problem is the Königsberg bridge problem: can you cross the seven bridges over the Pregel River without crossing over any bridge twice? Swiss mathematician Leonard Euler showed that the question was equivalent to the following problem: can you draw the graph of Fig. 2 without lifting pencil from paper, and without tracing any edge twice? Euler proved that it was impossible, and that any connected linear graph can be drawn with one stroke without retracing the edges if and only if the graph has either no odd vertices or just two odd vertices, where a vertice is odd if it is the endpoint of an odd number of lines. Because Fig. 2 has four odd vertices, it cannot be drawn by one continuous stroke without retracing lines. FIGURE 1 FIGURE 2

EARLY TOPOLOGY Continued… However, Fig. 3 has two odd vertices, so it is possible to draw that figure continuously without retracing edges. These are examples of early topographical questions, but these problems were the basis for modern topography, and therefore the questions and concepts therein are still explored in topography today.   FIGURE 3 FIGURE 2

EXAMPLES OF TOPOLOGY What topological similarities do the basketball and football have? What topological similarities do the iron and mug have?