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filter54321
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Homework Statement
What is the closure of F = { (x,sin(1/x) : 0<x<=1 }?
Homework Equations
None
The Attempt at a Solution
F is a squiggly line in R2. For every point in F (every point on the squiggly line) an open ball about that point will contain point both in F and in the complement of F. Therefore F is it's own boundary.
The closure of a set is equal to the unions of the boundary of the set and the set itself
Sclosure = dS U S
Therefore Fclosure = dF U F = F U F = F
My professor indicated that this line of reasoning is flawed. I'm not sure why nor am I sure what a correct anyswer would be. Any help would be appreciated.