A subset \(C\) of a topological space \(X\) is a closed set in \(X\) if and only if \(X\setminus C\) is open in \(X\text{.}\)
Theorem2.3.2.
Let \(X\) be a topological space. Then
\(\es\) and \(X\) are closed in \(X\text{;}\)
whenever \(A_\alpha\) is closed in \(X\) for all \(\alpha\text{,}\) then \(\cap_\alpha A_\alpha\) is closed in \(X\text{;}\) and
whenever \(A_1,\dots,A_n\) are closed in \(X\text{,}\) then \(\cup_{i=1n}A_i\) is closed in \(X\text{.}\)
Closed sets and continuity:
Theorem2.3.3.
Let \(X\) and \(Y\) be topological spaces and let \(f: X \to Y\text{.}\) Then \(f\) is continuous if and only for every closed set \(C\) in \(Y\text{,}\) the preimage \(f\inv(C)\) is closed in \(X\text{.}\)
Theorem2.3.4.CC14 = "Pasting Lemma".
Let \(X\) and \(Y\) be topological spaces, let \(A\) and \(B\) be closed subsets of \(X\text{,}\) and let \(A\) and \(B\) have the subspace topologies inherited from \(X\text{.}\) If \(f: X \to Y\) satisfies that \(f|A\) and \(f|B\) are both continuous, then \(f\) is continuous.
Example2.3.5.
Examples
Closed sets and constructions:
Theorem2.3.6.
If \(A\) is a subspace of \(X\) and \(C\) is a subset of \(A\text{,}\) then \(C\) is closed in \(A\) if and only if \(C = A \cap D\) for some closed set \(D\) in \(X\text{.}\)
Theorem2.3.7.
If \(A\) is closed in \(X\) and \(B\) is closed in \(Y\text{,}\) then \(A \times B\) is closed in \(X \times Y\text{.}\)
Subsection2.3.2Closures, interiors, boundaries, and limit points
Definition2.3.8.
Let \(X\) be a topological space and let \(A \sse X\text{.}\)
The closure of \(A\) in \(X\) is the set \(\text{Cl}_X(A) := \cap A \sse C \sse X, C \text{ closed in }X C\text{.}\)
The interior of \(A\) in \(X\) is the set \(IntX(A) := \cup U \sse A, U \text{ open in }X U\text{.}\)
The boundary of \(A\) in \(X\) is \(BdX(A) := \text{Cl}_X(A) \cap \text{Cl}_X(X-A)\text{.}\)
Example2.3.9.
Examples
Theorem2.3.10.
Let \(X\) be a topological space and let \(A\) and \(B\) be subsets of \(X\text{.}\)
\(\text{Cl}_X(A)\) is closed in \(X\) and \(IntX(A)\) is open in \(X\text{.}\)
If A \sse B \sse X, then \text{Cl}_X(A) \sse \text{Cl}_X(B) and IntX(A) \sse IntX(B).
\(A\) is closed in \(X\) iff \(A = \text{Cl}_X(A)\) and \(B\) is open in \(X\) iff \(B = IntX(B)\text{.}\)
\(\text{Cl}_X(A)\) is the smallest closed set in \(X\) containing \(A\text{,}\) and \(IntX(A)\) is the largest open set in \(X\) contained in \(A\text{.}\)
Interaction with continuity:
Theorem2.3.11.
If \(X\) and \(Y\) are topological spaces and \(f: X \to Y\text{,}\) then \(f\) is continuous if and only if for all \(A \sse X\text{,}\)\(f(\text{Cl}_X(A)) \sse \text{Cl}_Y(f(A))\text{.}\)
Interactions with constructions:
Theorem2.3.12.
Let \(Y\) be a subspace of \(X\text{,}\) and let \(A\) be a subset of \(Y\text{.}\) Then \(\text{Cl}_Y(A) = \text{Cl}_X(A) \cap Y\text{.}\)
Theorem2.3.13.
Let \(X_\alpha\) be a topological space and let \(A_\alpha \sse X_\alpha\) for all \(\alpha\text{.}\) Let \(\prod_\alpha X_\alpha\) have the product topology. Then \(Cl\prod_\alpha X_\alpha (\prod_\alpha A_\alpha ) = \prod_\alpha ClX_\alpha (A_\alpha )\text{.}\)
Theorem2.3.14.
Let \((X,\cT )\) be a topological space, let \(A \sse X\text{,}\) and let \(p \in X\text{.}\)
The point \(p \in \text{Cl}_X(A)\) if and only if every open set in \(X\) that contains \(p\) intersects \(A\text{.}\)
If \(\cT = \cT (\cB )\) is the topology generated by a basis \(\cB\text{,}\) then \(p \in \text{Cl}_X(A)\) if and only if every basis element containing \(p\) intersects \(A\text{.}\)
Definition2.3.15.
Let \(X\) be a topological space, let \(A \sse X\text{,}\) and let \(p \in X\text{.}\) The point \(p\) is a limit point of \(A\) in \(X\) if and only if for every open set \(U\) in \(X\) containing \(p\text{,}\) the set \((A -\{p\}) \cap U \ne \es\text{.}\) The set of limit points of \(A\) in \(X\) is denoted \(LimX(A)\text{.}\)
Example2.3.16.
Examples
Theorem2.3.17.
Let \(X\) be a topological space, let \(A \sse X\text{,}\) and let \(p \in X\text{.}\) Then \(p \in LimX(A)\) if and only if \(p \in \text{Cl}_X(A\setminus \{p\})\text{.}\)
Theorem2.3.18.
Let \(X\) be a topological space and let \(A \sse X\text{.}\) Then \(\text{Cl}_X(A) = A \cup LimX(A)\text{.}\)
