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lavinia
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yes. whenever one says Rn one means the usual topology.Ravi Mohan said:Thanks for explaining that. Now regarding my second question
yes. whenever one says Rn one means the usual topology.Ravi Mohan said:Thanks for explaining that. Now regarding my second question
DrGreg said:If ##\theta## was a member of the real line ##\mathbb{R}##, your objection would be valid. But it's not, it's a member of ##S^1##, "real numbers mod 2##\pi##", so "##\theta = 0##" and "##\theta = 2 \pi##" are two descriptions of the same point in ##S^1##, and of the same point in ##\mathbb{R}^2## under stevendaryl's mapping (for a given ##z##).
Cruz Martinez said:I think you're repeating some of the misconceptions you had on a previous thread here.
Cruz Martinez said:All manifolds are closed
PeterDonis said:Doesn't a closed manifold have to be compact? .
In which sense do you mean closed?PeterDonis said:Not sure what previous thread you're referring to (possibly you have confused me with another poster?), but see my response to DrGreg.
Doesn't a closed manifold have to be compact? Not all manifolds are compact.
lavinia said:a closed manifold can not be covered by a single chart since no open subset of Rn is compact, So the circle can not be covered by a single chart - nor can a sphere or a doughnut.
- a manifold my not be closed even as a metric space.
Cruz Martinez said:In which sense do you mean closed?
You said that every manifold is closed. But whatever definition of closed you choose this is not true. The plane minus a point is not closed as a metric space and is not a closed manifold.Cruz Martinez said:Which is exctly why i asked in which sense he means closed this time, because last time i saw he used it in a very different (wrong) way. With this of course i agree that the sphere cannot be covered by one chart since homeos preserve compactness as all continuous maps do.
lavinia said:You said that every manifold is closed. But whatever definition of closed you choose this is not true. The plane minus a point is not closed as a metric space and is not a closed manifold.