>>11950467You've likely studied Euclidean geometry in school, so you know how to draw triangles, etc. on a flat piece of paper.
Let's take "space" to mean anything with a number of points. The Euclidean plane (your paper) is a "space". The 3D room around you is a "space". Looking at it abstractly, the surface of the Earth is a "space".
If you look at the surface of a sphere, it's definitely not a Euclidean space: In Euclidean geometry, the sum of every angle in a triangle is 180° which is not true for the surface of a sphere.
However, if you only look at a small patch of the sphere, it is approximately true. For instance, you perceive the Earth as flat if you look from above in space.
A manifold is every "space" with this property: Locally, it looks like a Euclidean plane.
A 2D circle is a manifold (it looks like a straight line if you look at it from the side), a sphere (it looks like a plane from one side), your room (it looks like a 3D-Euclidean space), etc.
The cool thing about manifolds is that it's possible to describe them completely using only Euclidean spaces. Since we know Euclidean space very well, that's a good thing.
Ex: A map of England is a good way of describing England, although it really is part of a round object. You can patch a lot of maps together to get a whole atlas covering the Earth, which gives you a nice description of the 3D Earth using only 2D planes.
And that's a manifold. It's a space where you can create an atlas of charts, each of which is a part of a Euclidean space describing a part of the space.
