Topology Marianelly Lopez Anita Roman Jose Valencia Joe Williams

Topology vs. Geometry Topology is the study of objects/spaces that are the same under homeomorphism. In topology distance or angles of an object/ space doesn’t matter. In geometry angles or distances are required to find the shape of an object.

Dimensions 0-D: points 1-D: object that can be disconnected by removing a finite # of 0-D objects 2-D:object that can be disconnected by removing a finite # of 1-D objects 3-D: object that can be disconnected by removing a finite # of 2-D objects

Cuts Used to determine dimension Definition of Dimension Can be used a finite number of times Cannot be used to make homeomorphism true for shapes

Graphs Trees- shapes that force you to take the same path to reach the starting point Cycles- shapes that you can take another path around to reach the starting point

Homeomorphism Homeomorphism are continuous invertible functions/ maps.Homeomorphism are continuous invertible functions/ maps. Continuous functions/maps are two points that start close to each other and end close to each other.Continuous functions/maps are two points that start close to each other and end close to each other.

The Euler Characteristic χ= v - e Χ= kai, v= vertices, e= edges Χ= kai, v= vertices, e= edges Theory for trees χ=1 Theory for cycles χ= 0 Boundary points cannot be in the middle of a line in a homeomorphic shape

Mobius Strip A non-oriental surface that has only one boundary. When something goes from one end of it all the way around, they completely flip upside down. “P=d”