0.1. Introduction - What is Topology?

From a Talk by Artur Gorka

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The Intermediate Value Theorem

Theorem (Geometry):

Let and be any two points in the plane such that , then the segment from to crosses the .

Proof:

All points on the segment are of the form with

Hence we need to find such a such that But since

and

Thus is on the segment from to .

Theorem (Topology):

Let be any continuous function such that and then there exists some

$\QTR{Large}{0<x<1}$ such that

Proof: Forthcoming.

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The Jordan Curve Theorem

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The Seven Bridges of Konigsberg Over the River Pregel:

From A History of Topology

One asks whether or not there a path over the seven bridges that only traverses each bridge once.

It almost goes without saying that the answer to this question has nothing to do with the length of the bridges or the size of the islands. In fact, that no such path exists can be seen from a combinatorial argument rather than an argument about continuity.

To see this, consider the associated Graph:

and argue about traversal properties of nodes attached to an even then an odd number of edges.

In a "one-time traverse"

if a node has an even number of edges - a path must both begin and end there, or neither begin nor ends there .
if a node has an odd number of edges - a path must begin there or end there but not both.

Hence, a one-time traverse can have at most two nodes with an odd number of edges (due to Euler).

Since the Seven Bridges of Konigsberg Over the River Pregel problem involves four "nodes" with an odd number of edges, there can not be a one-time traverse.

Another Point of View (Arranging Dominos):

Can we arrange this set of dominos so that they all touch and that touching squares have the same number of dots?

Counting Holes - An Informal Discussion of Euler's Formula:

A Connected 1-complex is a connected finite set of line segments, called edges, in the plane that may share common end-points, called vertices, but do not intersect in their interiors.

A 1-complex may or may not divide its complement in the plane into separate regions. Here are some examples:

In fact, we can count the number of regions by simply subtracting the number of vertices from the number of edges and adding $\QTR{Large}{2}$ . $\QTR{Large}{\ }$