Practice Topology Intuition in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
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.
A coffee mug and a donut are 'the same' topologically—both have one hole.
Showing a random 20 of 50 problems.
Example 1
mediumHow many holes does the letter 'B' (printed) have, topologically?
Example 2
hardWhat is the Euler characteristic of a triple torus (3 holes)?
Example 3
easyHow many holes does a coffee mug have?
Example 4
hardFor an octahedron, , , . Verify Euler's formula.
Example 5
mediumWhat is the Euler characteristic of a torus?
Example 6
challengeA pretzel-like surface has 3 holes. What is its genus, and what is its Euler characteristic (using )?
Example 7
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.
Example 8
easyIs a figure-8 topologically equivalent to a circle?
Example 9
challengeThe Königsberg bridges problem asks whether one can traverse all bridges exactly once. Euler proved it impossible. What graph property guarantees a solution?
Example 10
easyTopologically, how many holes does a sphere have?
Example 11
easyHow many holes does a donut (torus) have?
Example 12
mediumAre the numerals '8' and '0' topologically the same?
Example 13
mediumThe number of holes in a surface has a name. What is it?
Example 14
easyIs a circle topologically the same as a figure-8?
Example 15
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'.
Example 16
mediumWhy are a coffee mug and a donut topologically the same but a coffee mug and a bowl are not?
Example 17
mediumTopologically, group these letters by number of holes: A, C, O, D.
Example 18
hardFor a double-torus (pretzel with two holes), what is its Euler characteristic?
Example 19
challengeExplain why topology is sometimes called 'rubber-sheet geometry' and what property survives all the stretching.
Example 20
mediumAre two linked rings topologically equivalent to two unlinked rings?