Open manifold, open set, and open cover

Question

According to https://en.wikipedia.org/wiki/Closed_manifold "In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components."

Closed manifold is a compact manifold without boundary.

Open manifold is a manifold without boundary that has only non-compact components.

Questions:

1. Is open manifold opposite or antonym of closed manifold? (It seems not, but why not?)

2. How do we call "compact manifold with boundary"?

3. What are the differences between open manifold and non-compact manifold?

4. How is the concept "open" in open manifold compared with the "open" in open set in set theory with topology, and open cover in topology? These open seem to mean different things.

Answer 1

The terminology is misleading, but this has little to do with the usual notion of closed / open in topology. Little, but not nothing:

• We refer to compact manifolds without boundary as closed manifolds since they are "closing in on themselves". Luckily enough, when embedded into other manifolds, they are always closed subsets by compactness.
• Compact manifolds with boundary are just called compact manifolds with boundary. I do not know of any widespread terminology for this.
• The terminology open manifolds is used in many different contexts. Some authors want them to be complete and non-compact Riemannian manifolds. Some others define them as manifolds that are not closed. Thus, it highly depends on the author and on the purpose of the subject. Unluckily, when embedded into some other manifolds, they are rarely open subsets, so the terminology is not that good and one has to be careful, for instance, by explicitly mentioning what is meant by that.