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I was doing some topology problems and well they were asking me to find the closure, interior and so on. The problem is not really asking for a justification on my part, but what if it did?

For example given $$A=\{(x,y)\; | \; x^2+y^2<1\}$$

if it asked me to find the closure I would just write this: $$\bar{A}=\{(x,y)\; | \; x^2+y^2 \leq 1\}$$

but what else could I add to justify the above.

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  • $\begingroup$ Thank you for the edit. I had forgotten to recheck it before I posted it. $\endgroup$
    – katari
    Nov 12 '12 at 3:35
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Remember the definition of closure of a given set. One of the definitions you can use is the definition that says a point is in the closure of a set if and only if you can take any neighborhood you want of this point (open balls centered in this point) you will always have no-empty intersections of this neighborhood and the set itself. Points of $A$ always are in the closure according this definition. it suffices to prove the circle $x^2+y^2=1$ is in the closure of $A$. You can prove this with simple geometry if you want to be formal, you have to prove that any neighborhood you take of any point on this circle you have always a no-empty intersection of this neighborhood and the circle.

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  • $\begingroup$ Thank you that was exactly what I was looking for. $\endgroup$
    – katari
    Nov 12 '12 at 3:23
  • $\begingroup$ you're welcome, you can upvote my answer also :) $\endgroup$
    – user42912
    Nov 12 '12 at 3:30

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