Topology Intuition Math Example 2
Follow the full solution, then compare it with the other examples linked below.
Example 2
hardA coffee mug and a donut (torus) are famously topologically equivalent. A sphere and a donut are not. Explain why, using the concept of holes.
Solution
- 1 Step 1: Define the topological property in question — the number of 'through-holes' (technically, genus) in a surface.
- 2 Step 2: Analyze the donut (torus). It has exactly one through-hole — the hole in the middle of the donut.
- 3 Step 3: Analyze the coffee mug. It has exactly one through-hole — the hole through the handle. The cup's cavity is not a through-hole; it is just an indentation.
- 4 Step 4: Analyze the sphere. A sphere (like a ball) has no through-holes — it is a closed surface with genus 0.
- 5 Step 5: Conclude: Mug and donut both have genus 1 (one through-hole), so they are topologically equivalent. A sphere has genus 0, so it is topologically different from both.
Answer
Mug ≅ Donut (both genus 1). Sphere ≇ Donut (sphere is genus 0).
In topology, the number of through-holes (genus) is a topological invariant — it cannot change under continuous deformation. A coffee mug has one through-hole (the handle), just like a donut, making them topologically equivalent. A sphere has no holes and is topologically distinct.
About Topology Intuition
Properties of shapes that are preserved under continuous deformation (stretching, bending, and twisting, but not tearing or gluing). Topology studies what remains the same when you treat shapes as if they were made of infinitely stretchable rubber.
Learn more about Topology Intuition →More Topology Intuition Examples
Example 1 medium
A rubber band is shaped like a circle. If you stretch and reshape it (without tearing or gluing), ca
Example 3 easyWhich pairs of shapes are topologically equivalent (same number of holes)? (a) Triangle and circle.
Example 4 mediumA topologist says that the number of times a closed curve crosses itself is a topological property t