# Tagged Questions

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### 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?
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### 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 $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
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### 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 ...
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### 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, ...
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### 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 ...