\part{Manifolds}
Informally, they capture the idea that some topological spaces 'look' Euclidean locally, but are possibly not Euclidean. For example, Earth's surface 'looks' like $\mathbb{R}^{2}$ from a human's point of view (ignoring bumps and mountains on the ground), but we know that the Earth is topologically a $2$-sphere; if you walk in a straight line for long enough, you eventually reach your starting point again. Alternatively, if one shines a light parallel with the ground, it will traject above a far enough observer due to the curvature of the earth; this is how we know the Earth isn't flat.

Manifolds are especially important in differential topology, which devotes itself to the studies of 'differentiable manifolds'.

- Manifold

- Jordan curve theorem $\mathbb{R}^2$
- Generalized Jordan curve theorem
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