= Geometry and topology
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How much information about a shape do we need to keep? <Euclidean geometry> retains <lengths> and <angles>, allowing questions about the diagonal of a <rectangle> or the shortest route between two points. <Topology> retains the structure preserved by a <homeomorphism>, allowing questions about <connectedness> and whether a loop can shrink to a point, as recorded by the <fundamental group>. These are different levels of description of the same object.
For example, stretching a <circle> horizontally produces an <ellipse>. The stretching map and its inverse are <continuous maps>, so it gives a <homeomorphism>: the two curves have the same <topology>. It changes some <Euclidean distances>, however, so it is not an <isometry>. This distinction helps separate essential structure from a particular drawing, while explaining why measuring a shape requires more information than recognizing its <topology>.
The choice of geometry matters too. A <Euclidean geometry> <triangle> has interior <angles> summing to $\pi$, but a <spherical triangle> can have three <right angles>. <Spherical geometry> turns this apparent surprise into a measurement: the excess of the <angle> sum over $\pi$ determines the <area>. Begin with the <Pythagorean theorem> for flat measurements, then compare the <spherical triangle> below to see <curvature> change the rules.
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