Topology

Study of geometric properties preserved under continuous deformations.

What is Topology?Open and Closed SetsContinuous Functions
Topological InvariantsCompactness and ConnectednessManifolds and Higher Dimensions
Computer Graphics and AnimationRobotics and Path PlanningData Analysis and Machine Learning
Visualize and DrawUse Analogies and Stories
What is Topology?Open and Closed SetsContinuous FunctionsTopological Invarian...Compactness and Conn...Manifolds and Higher...Computer Graphics an...Robotics and Path Pl...Data Analysis and Ma...Visualize and DrawUse Analogies and St...
Advanced Topics

Topological Invariants

What Stays the Same? Topological Invariants

Topological invariants are properties of shapes that remain unchanged under continuous deformations. They're like fingerprints for spaces—no matter how you stretch or twist them, these features don't change.

Common Invariants

  • Number of holes: A donut and a coffee cup both have one hole.
  • Connectedness: Whether a shape is in one piece or several.

Why They Matter

Invariants help mathematicians classify spaces and prove when two shapes are not topologically the same.

Key Formula

\[\chi = V - E + F\]

Examples

  • A sphere has zero holes; a torus (donut) has one hole.

  • Euler characteristic (\(\chi\)) tells us about the shape of polyhedra.

In a Nutshell

Topological invariants like the number of holes help us tell spaces apart.

Key Terms

Euler Characteristic
A number that helps classify surfaces, given by \(V - E + F\) for vertices, edges, and faces.
Genus
The number of holes in a surface.
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Compactness and Connectedness