Practice Topology Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Properties that are preserved under continuous deformation (stretching, not tearing).

A coffee mug and a donut are 'the same' topologically—both have one hole.

Example 1

medium
A rubber band is shaped like a circle. If you stretch and reshape it (without tearing or gluing), can it become a square? Can it become the number '8'? Explain using topological thinking.

Example 2

hard
A coffee mug and a donut (torus) are famously topologically equivalent. A sphere and a donut are not. Explain why, using the concept of holes.

Example 3

easy
Which 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'.

Example 4

medium
A topologist says that the number of times a closed curve crosses itself is a topological property that distinguishes curves. A circle crosses itself 0 times. A figure-8 crosses itself once. Can you continuously deform a figure-8 into a circle without lifting it from the plane?
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