Interior, Closure, Boundary
Topological operations on sets.
Discover Bridges
Study Interior, Closure, Boundary
Definition 1 (Interior, closure, boundary). For \(C\subset \mathbb{K}\), we define the
interior of \(C\) by the set \[\mathring{C}=\{x\in C\,:\,\text{ there exists some }\varepsilon>0\text{ such that }B_{\varepsilon}(x)\subset C\}.\] The elements of \(\mathring{C}\) are called inner points of \(C\).
closure of \(C\) by the set \[\overline{C}=\{x\in \mathbb{K}\,:\,\text{ there exists a sequ. }(a_n)_{n\in\mathbb{N}}\text{ in $C$ with }\lim_{n\to\infty}a_{n}=x\}.\] The elements of \(\overline{C}\) are called osculation points of \(C\).
boundary of \(C\) by the set \[\partial C=\overline{C}\backslash \mathring{C}.\] The elements of \(\partial{C}\) are called boundary points of \(C\).
Remark 2. The relation \(\mathring{C}\subset C\subset \overline{C}\)
holds true for arbitrary subsets \(C\subset
\mathbb{K}\). The first inclusion holds true by definition of
\(\mathring{C}\). To verify \(C\subset \overline{C}\) we take an
arbitrary \(x\in C\) and consider the
constant sequence \((x)_{n\in\mathbb{N}}\). Since this sequence
is completely contained in \(C\) and
converges to \(x\in C\), we must have
that \(x\in\overline{C}\).
It can be shown that for all sets \(C\), \(\mathring{C}\) is always open and \(\overline{C}, \partial C\) are always
closed sets. In particular, if \(C\) is
open (closed), then we have \(\mathring{C}=C\) (resp. \(\overline{C}=C\))
Example 3. These are examples in the case \(\mathbb{K} = \mathbb{R}\):
\(C=[0,1]\), then \(\mathring{C}=(0,1)\), \(\overline{C}=[0,1]\) and \(\partial C=\{0,1\}\);
\(C=\{\frac1n\,:\,n\in\mathbb{N}\}\), then \(\mathring{C}=\emptyset\) and \(\overline{C}=\partial C=\{0\}\cup \{\frac1n\,:\,n\in\mathbb{N}\}\).
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