Problems

The left figure is formed by two interlocking loops joined to a solid ball. The right figure is formed by two unlinked loops joined to a solid ball. Describe how to transform the left into the right without cutting, tearing or passing the loops through each other.

A pair of points on a circle are said to be antipodal if they are on two opposite ends of a common diameter. P and Q in the picture are antipodal points. If we glue every pair of antipodal points on a circle, then what is the resulting shape?

A surface P is created by gluing every pair of antipodal points of a disc (a circle with inside filled in). We represent P on the plane by a disc in the following picture and bear in mind that the antipodal points are glued.

Explain why the two diameters in the pictures are in fact two circles on P and how to stretch it so that it becomes a single loop not touching any of the glued points.

Today we will draw lots of pictures.

The subject is Topology. It is often called “rubber-sheet geometry" because while it is the study of shapes, topologists typically do not pay too much attention to rigid notions like angle and lengths. We have much more flexibility in topology. Some common words describing the operations here might include “gluing", “stretching", “twisting" and “inflating".

Although we will not define continuity, it is a more or less intuitive idea. Topological operations should be continuous. If you have a line segment, no amount of stretching, twisting or bending can make it into two disconnected segments.

We can define the absolute value \(|x|\) of any real number \(x\) as follows. \(|x|=x\) if \(x\ge0\) and \(|x|=-x\) if \(x<0\). What are \(|3|\), \(|-4.3|\) and \(|0|\)?

Prove that \(|x|\ge0\).

Prove that \(|x|\ge x\). It may be helpful to compare each of \(|3|\), \(|-4.3|\) and \(|0|\) with \(3\), \(-4.3\) and \(0\) respectively.

It is possible to play tic-tac-toe on a torus: gluing the sides means that the bottom row is above the top row and the right most column is also to the left of the left most column. Is one of the players guaranteed to win if they play all the right moves?

Let \(A=\{1,2,3\}\) and \(B=\{2,4\}\) be two sets containing natural numbers. Find the sets: \(A\cup B\), \(A\cap B\), \(A-B\), \(B-A\).

Let \(A=\{1,2,3,4,5\}\) and \(B=\{2,4,5,7\}\) be two sets containing natural numbers. Find the sets: \(A\cup B\), \(A\cap B\), \(A-B\), \(B-A\).

Given three sets \(A,B,C\). Prove that if we take a union \(A\cup B\) and intersect it with the set \(C\), we will get the same set as if we took a union of \(A\cap C\) and \(B\cap C\). Essentially, prove that \((A\cup B)\cap C = (A\cap C)\cup (B\cap C)\).