In Spivak's Diff Geom (vol.1), p.19, he says a closed manifold is non-bounded and compact (A point in boundary has a neighborhood homeomorphic to half-space). I don't know a non-trivial example of that.
For example, compact subset of $\mathbb{R}^2$ is usually closed set and has boundary, so it's not closed manifold according to Spivak's definition.
An example is the finite set of discrete points of $\mathbb{R}^2$, it's compact and, since no point in it has a neighborhood homeomorphic to half-space, it has no boundary. But the example is trivial.
Does anyone know a non-trivial example of closed manifold (in Spivak's definition)?
[Well I check the definition of closed and compact manifolds here:
https://mathworld.wolfram.com/ClosedManifold.html
https://mathworld.wolfram.com/CompactManifold.html
It seems what confuses me is that 'compact' here means $\sigma$-compact (locally compact and connected, or say its any open cover has countable sub cover), and I thought it means that its open cover has finite sub cover. Right?]
Best Answer
The standard first examples of closed manifolds are the spheres. For instance, $S^2$, usually imagined as the unit sphere in $\Bbb R^3$. The torus is also a popular example, and you will eventually be very familiar also with the projective plane and the Klein bottle. If you want to make your own examples, the boundary of compact manifolds with boundary will always work.
"Compact" here does not mean $\sigma$-compact. Indeed, $\sigma$-compactness is one of Spivak's requirements for any manifold. If I recall correctly, there is an appendix exploring a few non-$\sigma$-compact examples, like the so-called long line.