The Defining Property of Closed Sets
A set \( A \) is closed if and only if the closure of \( A \) is equal to \( A \) in the topological space. $$ A = \text{Cl}(A) $$
Practical Example
Consider the topological space \( \mathbb{R} \) with the standard topology and the set \( A = [0, 1] \).
A set is closed if it contains all of its limit points. The limit points of \( A = [0, 1] \) are all the points between \( 0 \) and \( 1 \), including the endpoints.
The set \( A \) clearly contains all these points, so \( A \) is a closed set.
Now, let's verify if \( A = \text{Cl}(A) \).
The closure of \( A \) in the standard topology is the set itself, \( \text{Cl}(A) = [0, 1] \), because \( [0, 1] \) contains all its limit points.
$$ A = \text{Cl}(A) $$
This example confirms that the set \( A = [0, 1] \) is closed because \( A \) coincides with its closure.
Additionally, it confirms that a set \( A \) is closed if and only if \( A = \text{Cl}(A) \).
The Proof
Let's start from the basics:
- Definition of closure: The closure of a set \( A \), denoted as \( \text{Cl}(A) \), is the set of all points in \( A \) plus its limit points. Formally: \[ \text{Cl}(A) = A \cup \{x \in X \mid \text{ every neighborhood of x contains a point of A } \} \]
- Closed set: A set \( A \) is defined as closed if it contains all its limit points. Thus, \( A \) is closed if and only if \( A = \text{Cl}(A) \).
We need to prove the implication in both directions:
1] If \( A \) is closed, then \( A = \text{Cl}(A) \):
If \( A \) is closed, by definition, it contains all its limit points.
Therefore, there are no limit points of \( A \) that are not already in \( A \).
Since the closure of \( A \) is the union of \( A \) with its limit points, we have:
$$ \text{Cl}(A) = A \cup \{ \text{limit points of A} \} = A $$
Thus, \( A = \text{Cl}(A) \).
2] If \( A = \text{Cl}(A) \), then \( A \) is closed:
If \( A = \text{Cl}(A) \), it means that \( A \) contains all its limit points, as \( \text{Cl}(A) \) includes all the points of \( A \) plus its limit points.
Therefore, by definition, \( A \) is closed.
