Closed Sets#
Definition: Closed Set
A subset \(S \subseteq X\) of a topological space \(X\) is closed if each point in its complement \(X \setminus S\) has at least one neighborhood which is disjoint from \(S\):
\[\forall p \in X \setminus S: \exists N(p) \text{ with } N(p) \cap S = \varnothing\]
Theorem: The Fundamental Properties of Closed Sets
If \(X\) is a topological space, then:
- The empty set \(\varnothing\) and \(X\) itself are closed.
- If \(\mathcal{S}\) is a collection of closed sets, then its intersection is also closed.
- If \(\mathcal{S}\) is a finite collection of closed sets, then its union is also closed.
Proof
We need to prove three things:
- (I) The empty set \(\varnothing\) and \(X\) itself are closed.
- (II) If \(\mathcal{S}\) is a collection of closed sets, then its intersection is also closed.
- (III) If \(\mathcal{S}\) is a finite collection of closed sets, then its union is also closed.
Proof of (I):
Proof of (II):
Proof of (III):
Theorem: Closed Sets via Open Sets
Let \(X\) be a topological space.
A subset \(S \subseteq X\) is closed if and only if its complement \(X \setminus S\) is open.
Proof
TODO
Theorem: Closed Sets via Closure
A subset of topological space is closed if and only if it is equal to its own closure:
\[S = \overline{S}\]
Proof
TODO