Union of a Set's Boundary and Interior
The union of a set's boundary \( \partial A \) and its interior \( Int(A) \) is equal to the closure of \( A \). $$ \partial A \cup \text{Int}(A) = \text{Cl}(A) $$
An Example
Let's consider the example where \(A = (0, 1)\) in the topological space \(\mathbb{R}\):
The interior of \(A\) is the open interval (0,1).
$$ Int(A) = (0, 1) $$
The closure of \(A\) is the closed interval [0,1], including the endpoints.
$$ Cl(A) = [0, 1] $$
The boundary of \(A\) consists of the points 0 and 1.
$$ \partial A = \{0, 1\} $$
The union of \(A\)'s boundary and interior is precisely its closure:
$$ \partial A \cup Int(A) = \{0, 1\} \cup (0, 1) = [0, 1] $$
$$ \partial A \cup Int(A) = Cl(A) $$
This property illustrates that all points in \(A\), including its boundary points, are encompassed by the union of its interior and boundary.
Proof
To formally prove this property, let's review some definitions:
- Interior of \(A\) (\(Int(A)\))
The interior of \(A\) is the set of all points in \(A\) that have a neighborhood entirely contained within \(A\). - Closure of \(A\) (\(Cl(A)\))
The closure of \(A\) is the set of all points in \(A\) plus its boundary points. In other words, \(Cl(A) = A \cup \partial A\). - Boundary of \(A\) (\(\partial A\))
The boundary of \(A\) is the set of points that belong to both the closure of \(A\) and the closure of \(A\)'s complement. Formally, \(\partial A = Cl(A) \cap Cl(X - A)\).
Consider the set \(A\) in a topological space \(X\).
By definition, the closure of \(A\) is:
$$ Cl(A) = Int(A) \cup \partial A $$
The interior of \(A\) is, by definition, disjoint from the boundary of \(A\):
$$ Int(A) \cap \partial A = \emptyset $$
Therefore, we can write:
$$ Cl(A) = Int(A) \cup \partial A $$
And so forth.
