Simple explanations of differences between topology and geometry

Question

I have google but still can't find a simple explanations of differences between topology and geometry. Most of answers on internet are not easy to understand for non-math major. Based on what I found (these might be incorrect):

One of papers: roughly states the following: Topology focuses on the '2D' structure but Geometry focuses on the '3D' structure and spatial relations.

https://arxiv.org/pdf/2110.07728.pdf

Can anyone kindly provide simple explanations and examples to NON-math major audiences?

Thanks in advance!

Answer 1

The main difference is that geometry carries the word metric. It requires a measure of lengths, distances, and angles. Classical geometry is flat, in whatever dimension, and non-classical geometry is allowed to have curved spaces, e.g. the geometry on the earth's surface, or in local spacetime. The sum of all angles in a triangle is 180° in a flat (Euclidean) geometry, but greater than 180° on earth, and there are geometries where it is less than 180°.

Topology is the theory of continuous functions between spaces (roughly said, those functions which can be drawn in one line, i.e. without gaps, but this is really meant only as a heuristic). The spaces can have but do not require a metric that measures distances. All it needs is a concept of points, sets of points, and a formal definition of what nearby means. This is necessary to formally describe "drawn in one line, i.e. without gaps". Gaps are not nearby.

A ring and a mug are two very different things in geometry but are the same thing in topology where only the hole counts, not the form.

Answer 2

I would say that topology is part of geometry. Geometry is the study of "spaces" and certain collections of mappings between them. For instance classical geometry is the study of Euclidean space and the isometries of Euclidean space. Differential geometry is the study of smooth manifolds and differentiable maps between them. Complex geometry is the study of complex manifolds and the holomorphic maps between them. Riemannian geometry is the study of manifolds with a notion of distance, and the collection of differentiable maps between these. Algebraic geometry is the study of spaces which are zero sets of polynomials, and the maps between them are required to be algebraic in some sense.

Topology is a very "loose" kind of geometry where we only require our spaces to have a topological structure on them (a notion of "nearness") and the maps we study are continuous maps (they sort of "preserve nearness").