Topology

Study of geometric properties preserved under continuous deformations.

What is Topology?Types of Topological SpacesOpen and Closed Sets
Homeomorphisms and Topological EquivalenceConnectedness and CompactnessTopological Invariants
Topology in Computer ScienceTopology in Biology and MedicineEveryday Uses of Topology
Visual Learning with DiagramsActive Recall and Practice Problems
What is Topology?Types of Topological...Open and Closed SetsHomeomorphisms and T...Connectedness and Co...Topological Invarian...Topology in Computer...Topology in Biology ...Everyday Uses of Top...Visual Learning with...Active Recall and Pr...
Advanced Topics

Homeomorphisms and Topological Equivalence

Homeomorphisms: The Heart of Topology

A homeomorphism is a special function that shows when two spaces are “topologically the same.” It is a continuous, one-to-one, and onto mapping with a continuous inverse.

What Does This Really Mean?

If you can stretch or bend one object into another without cutting or gluing, they are homeomorphic.

Applying Homeomorphisms

Homeomorphisms help classify spaces and solve problems in mathematics, physics, and engineering.

Examples in Action

  • Transforming a clay donut into a clay coffee cup with a handle.
  • Morphing a rubber band into a circle or an ellipse.

Examples

  • A loop of string and a perfect circle are homeomorphic.

  • A square and a triangle (with flexible sides) are homeomorphic.

In a Nutshell

Homeomorphisms reveal when two objects are fundamentally the same in topology.

Key Terms

Continuous function
A function where small changes in input produce small changes in output.
Inverse
A function that reverses the effect of another function.
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Connectedness and Compactness