9.7. Compactness¶
Most important topological property
If we don’t understand compactness, we don’t understand topology
Compactness is topological generalization of finiteness
9.7.1. Compactness¶
Definition
Let \(A\) be a subset of a topological space \((X, \mathcal{T})\). Then \(A\) is said to be compact if for every set \(I\) and every family of open sets, \(\{O_i, i \in I | A \subseteq \cup_{i \in I}O_i\}\) there exists a finite subfamily \(\{O_{i_1},O_{i_2},\dots, O_{i_n}\}\) such that \(A \subseteq O_{i_1} \cup O_{i_2} \cup \dots \cup O_{i_n}\).
Example
If \((X, \mathcal{T}) = \mathbb{R}\) and \(A = (0, \infty)\) then \(A\) is not compact.
Let \((X, \mathcal{T})\) be a topological space and \(A = \{x_1, x_2, \dots, x_n\}\) be any finite subset of \((X, \mathcal{T})\). Then \(A\) is compact.
Remark
Every finite set in a topological space is compact. “Compactness” is a generalization of “finiteness”.
Example
A subset \(A\) of a discrete space \((X, \mathcal{T})\) is compact if and only if it is finite.
9.7.1.1. Cover¶
Definitions
Let \(I\) be a set and \(O_i, i \in I\) a family of open sets in a topological space \((X, \mathcal{T})\). Let \(A\) be a subset of \((X, \mathcal{T})\). Then \(O_i, i \in I\) is said to be an open covering of \(A\) if \(A \subseteq \cup_{i \in I} O_i\). A finite subfamily, \(O_{i_1}, O_{i_2}, \dots, O_{i_n}\) of \(O_i, i \in I\) is said to be a finite subcovering (of \(A\)) if \(A \subseteq O_{i_1} \cup O_{i_2} \cup \dots \cup O_{i_n}\).
Now we can rephrase compactness
Definition
A subset \(A\) of a topological space \((X, \mathcal{T})\) is said to be compact if every open covering of \(A\) has a finite subcovering. If the compact subset \(A\) equals \(X\), then \((X, \mathcal{T})\) is said to be a compact space.
Proposition
The closed interval \([0,1]\) is compact.
9.7.2. Heine-Borel Theorem¶
Proposition
Let \(f: (X,\mathcal{T}) \to (Y,\mathcal{T}_1)\) be a continuous surjective map. If \((X,\mathcal{T})\) is compact then \((Y,\mathcal{T}_1)\) is compact.
Corollary
Let \((X,\mathcal{T})\) and \((Y,\mathcal{T}_1)\) be homeomorphic topological spaces. If \((X,\mathcal{T})\) is compact then \((Y,\mathcal{T}_1)\) is compact.
Corollary
For \(a,b \in \mathbb{R}, a<b\), \([a,b]\) is compact while \((a,b)\) is not compact.
\([a,b]\) is homeomorphic to \([0,1]\) (a compact spacce).
\((a,b)\) is homeomorphic to \((0,\infty)\) (a non-compact space).
Proposition
Every closed subset of a compact space is compact.
Proposition
A compact subset of a Hausdorff topological space is closed.
Corollary
A compact subset of a metrizable space is closed.
\((a,b]\) and \([a,b)\) are not compact since they are not closed.
Proposition
A compact subset of \(\mathbb{R}\) is bounded.
Theorem (Heine-Borel Theorem)
Every closed bounded subset of \(\mathbb{R}\) is compact.
If \(A\) is a closed and bounded subset of \(\mathbb{R}\) then \(A \subseteq [a,b]\) for some \(a,b\in \mathbb{R}\).
\([a,b]\) is compact and \(A\) is a closed subset of a compact set, hence compact.
Proposition (Converse of Heine-Borel Theorem)
Every compact subset of \(\mathbb{R}\) is closed and bounded.
9.7.2.1. Metric spaces¶
Definition
A subset \(A\) of a metric space \((X,d)\) is said to be bounded if there exists a real number \(r\) such that \(d(a_1, a_2) \leq r \forall a_1, a_2 \in A\).
Proposition
Let \(A\) be a compact subset of a metric space \((X,d)\). Then \(A\) is closed and bounded.
Theorem (Generalized Heine-Borel theorem)
A subset of \(\mathbb{R}^n\) is compact if and only if it is closed and bounded.
Proposition
Let \((X,\mathcal{T})\) be a compact space and \(f\) a continuous mapping from \((X,\mathcal{T})\) into \(\mathbb{R}\). Then the set \(f(X)\) has a greatest element and a least element.
Proposition
Let \(a, b \in \mathbb{R}\) and \(f\) a continuous function from \([a,b]\) into \(\mathbb{R}\). Then \(f([a,b]) =[c,d]\) for some \(c, d \in \mathbb{R}\).