# Unit 4

## Topology's Twists and Turns

Hyperbolic Purse

Topology, known as "rubber sheet math," is a field of mathematics that concerns those properties of an object that remain the same even when the object is stretched and squashed. In this unit we investigate topology's seminal relationship to network theory, the study of connectedness, and its critical function in understanding the shape of the universe in which we live.

## Unit Goals

• Topology is the study of fundamental shape.
• Objects are topologically equivalent if they can be continuously deformed into one another.  Properties that are preserved during this process are called topological invariants.
• Intrinsic topology is the study of a surface or manifold from the perspective of being on or in it.
• Extrinsic topology is concerned with properties of a surface or manifold seen from an external viewpoint.  This requires some kind of embedding.
• The Euler characteristic is a topological invariant.
• Orientability is a topological invariant.
• A configuration space is a topological object that can be used to study the allowable states of a given system.
• The question of the shape of our universe is a question of intrinsic topology.

# Video Transcript

Can you imagine the shape of the universe? That's where Topology comes in: a branch of mathematics concerned with the study of spatial relationships that don't depend on measurement, and is more concerned with concepts like 'between' or 'inside,' and how things are connected.

# Textbook

Topology, originally known as analysis situs—roughly, "geometry of position", seeks to describe what is fundamental about shape in general.

# Interactive

Because this interactive covers an abstract concept of configuration space, you may benefit by reading the text chapter first. In this interactive you will learn what a configuration space is and how to model it as a surface.