Constructing Homeomorphisms

Topology · Axiom Academy

LESSON Constructing Homeomorphisms Techniques and tools for building explicit homeomorphisms between topological spaces Step 1: The Art of Construction While proving that two spaces are not homeomorphic can be done using invariants (like compactness or connectedness), proving they are homeomorphic requires us to explicitly construct a homeomorphism. This is both an art and a science, requiring creativity and a toolkit of standard techniques. A function between topological spaces is a homeomorphism if: is bijective (one-to-one and onto) In this lesson, we'll explore several powerful techniques for constructing homeomorphisms, from simple linear maps to sophisticated transformations like stereographic projection. Step 2: Linear Maps for Intervals The simplest homeomorphisms are linear maps between open intervals. Any two open intervals in are homeomorphic. This is a homeomorphism with inverse: Solution: Define . We can verify: This is a line with positive slope, so it's continuous and strictly increasing Step 3: Stereographic Projection One of the most beautiful and important homeomorphisms in topology is stereographic projection, which shows that a sphere with one point removed is homeomorphic to Euclidean space. For any , we have , where is the unit sphere in . The homeomorphism is given by projecting from the north pole onto the equatorial plane . For , we have . The stereographic projection from is: Step 4: Squaring and Exponential Maps

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