Continuous Maps and Homeomorphisms: Explained with Examples

A continuous map between topological spaces is one where the preimage of every open set is open: it never tears the space, though it may stretch or fold it. A homeomorphism is a bijective continuous map with continuous inverse, making two spaces topologically identical, the same shape up to stretching, like a coffee mug and a doughnut.

This defines what topology studies: properties preserved by homeomorphisms, called topological invariants. Connectedness, compactness and hole counts are invariants; distances and angles are not. Classifying shapes up to homeomorphism is topology's central business, from knots to the shape of the universe.

Key Concepts

• Continuous map: f: X → Y with f^{-1}(V) open in X for every open V in Y; nearby points stay nearby.
• Homeomorphism: bijective, with f and f^{-1} continuous; write X ≅ Y.
• Topological invariant: preserved by homeomorphisms, e.g. connectedness, compactness.
• Embedding: a homeomorphism onto its image; one space sitting nicely inside another.
• Composition: continuous maps compose to continuous maps; homeomorphisms to homeomorphisms.

Formulas

• Continuity test: f is continuous iff preimages of (sub)basis elements are open; equivalent to epsilon-delta in metric spaces.
• Compact-Hausdorff bonus: a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism.
• Invariance: X ≅ Y implies X connected iff Y connected, X compact iff Y compact.

Worked Example

(0, 1) is homeomorphic to R, but [0, 1) is not homeomorphic to (0, 1).

1. f(x) = tan(pi(x - 1/2)) maps (0,1) onto R, continuous and strictly increasing, hence bijective.
2. Inverse f^{-1}(y) = arctan(y)/pi + 1/2 is continuous. So f is a homeomorphism: (0,1) ≅ R.
3. Suppose g: [0,1) → (0,1) were a homeomorphism. Deleting 0 from [0,1) leaves (0,1), connected.
4. Deleting g(0) from (0,1) splits it into two disjoint nonempty open pieces: disconnected.
5. Connectedness is invariant, contradiction. The endpoint is topologically detectable.

Common Misconceptions

• Continuous bijections need not be homeomorphisms: t ↦ (cos 2 pi t, sin 2 pi t) from [0,1) onto the circle is bijective and continuous, but its inverse tears at the join.
• Homeomorphic spaces can look different: a square and a circle, or bounded (0,1) and unbounded R, are homeomorphic; boundedness is not invariant.
• "Unbroken pen stroke" is not the definition; the preimage condition is, agreeing with intuition in familiar spaces.

Real-World Applications

• Shape recognition: vision systems classify objects up to stretching homeomorphisms.
• Knot theory: knots studied up to homeomorphism; enzymes unknot DNA this way.
• Robotics: homeomorphic configuration spaces share motion-planning strategies.

Continuous maps preserve closeness through preimages of opens; homeomorphisms, continuous bijections with continuous inverses, identify spaces of the same topological shape. (0,1) ≅ R via the tangent map, while [0,1) differs from (0,1) by what point-deletion does to connectedness. Invariants are the tools that tell spaces apart.