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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

S

_{closure}= dS U S

Therefore F

_{closure}= 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.