1
vote
0answers
34 views

Why are compact and noncompact manifolds without boundary called closed manifolds and open manifolds, respectively?

Why not just call them compact and noncompact manifolds? Isn't the general assumption that manifolds have empty boundary unless stated otherwise?
2
votes
1answer
101 views

Why functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ are called cocycles?

Let $X$ be some smooth manifold and $\{U_\alpha\}$ be its open cover. The last month I hear very often that one calls a collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
1
vote
1answer
83 views

Difference between “Live” and “Define”

In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used. I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
3
votes
3answers
197 views

Class of manifolds is a set?

Is the class of all 2-countable manifolds a set? I think so: each such space is a countable union of sets of cardinality $|\mathbb{R}^n|\!=\!|\mathbb{R}|$, i.e. a manifold has cardinality continuum, ...
3
votes
2answers
339 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
0
votes
0answers
35 views

Regular Prototype

I have task: Can such smooth function no Möbius band exist to be satisfy follow condition: Central circle of Möbius band is "regular prototype" of some point. Can you say me, what means "regular ...
1
vote
1answer
98 views

What's the name of a submanifold-plus-any-missing-boundary?

Is there a standard name for the closure of a submanifold of some fixed manifold M? Example. The closed interval [0, 1] is not a manifold, because there is no atlas which contains charts at ...