Topology Intuition Math Example 3
Follow the full solution, then compare it with the other examples linked below.
Example 3
easyWhich pairs of shapes are topologically equivalent (same number of holes)? (a) Triangle and circle. (b) Letter 'O' and letter 'D'. (c) Letter 'B' and number '8'.
Solution
- 1 Step 1: Count holes in each shape. A triangle: 0 holes. A circle: 0 holes. → (a) equivalent.
- 2 Step 2: Letter 'O': 1 hole. Letter 'D': 1 hole. → (b) equivalent.
- 3 Step 3: Letter 'B': 2 holes (two enclosed regions). Number '8': 2 holes. → (c) equivalent.
Answer
All three pairs are topologically equivalent.
Topological equivalence among 2D letter shapes is determined by counting enclosed regions (holes). A triangle and circle both enclose no holes in their outline. 'O' and 'D' each have one enclosed region. 'B' and '8' each have two enclosed regions.
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 2 hardA coffee mug and a donut (torus) are famously topologically equivalent. A sphere and a donut are not
Example 4 mediumA topologist says that the number of times a closed curve crosses itself is a topological property t