Besides the other answers, things also start to get interesting when you add some qualifiers.

E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.

Good point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?

To be honest it's not even my field so I know a lot less than I wish I did but yeah agreed it has no shortage of interesting things to talk about :)

Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it.

Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.

> Good point

Can a point be a manifold?

Yes, but it's a simple one of dimension zero. There's not much to do here---every 'neighbourhood' of the point is exactly that: just the point.

(not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)

A Y-shaped "line".

Simplest example of what not a manifold is the shape of the letter “Y”. Technically for a topological space to be a manifold, every small neighborhood around each point should look the same. But “Y” has a special point at the center.

A fractal isnt a manifold. Discrete sets are an edge case. They are considered to be 0 dimensional manifolds to make them fit, but there is a not much that manifold theory has to say about them.

> A fractal isnt a manifold

Some are. A Koch curve for instance.

I believe you are correct. Koch curve is a continuous curve so its topological dimension would be 1 (not its fractal dimension).

Things with edges, like disks and finite lines.

Both of those are typically referred to as manifolds with boundary. The setting of Stokes Theorem is often manifolds with boundary.

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).

I thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.

Nothing wrong with them, they can be manifolds.

f(x) = 0 (x < 1, x > 1)

f(0) = 1