Different definitions of boundary
Wikipedia gives three apparently equivalent definitions of a boundary:
- the closure of S without the interior of $S: ∂S = \bar S$ \ $S^o$.
- the intersection of the closure of S with the closure of its complement: $∂S = \bar S ∩ \overline {(X \ S)}$.
- the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S.
The first two are equivalent because of something like this.
But how do I show the first two follow from the third? I don't even a rigorous symbolic definition to use in such a proof.
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$\begingroup$Suppose $p$ is such that every neighborhood of $p$ contains at least one point of $S$, and at least one point not of $S$.
Then, by the first property (every neighborhood of $p$ contains at least one point of $S$), $p$ is a limit point of $S$, so $p \in \overline S$.
The second property shows that $p$ cannot be in the interior of $S$. Thus the third definition implies the first. (It also directly implies $p$ is in the closure of the complement of $S$, if you are shooting for demonstrating the second definition holds: "$p$ is a limit point of complement of $S$" $\Leftrightarrow$ "every neighborhood of $p$ contains a point not in $S$" $\Leftrightarrow$ "there does not exist a neighborhood of $p$ contained in $S$" ).
Basically these same words in a different order will show that the first definition implies the third. (and getting to and from the second/third definitions proceeds in a similar manner).
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