Theorem. \(X\) is compact if and only if every net in \(X\) has a convergent subnet.
[Hint: To prove the implication \(\Rightarrow\), let \(B_\alpha = \ts{x_\beta \mid \alpha \preceq \beta}\) and show that \(\ts{B_\alpha}\) has the finite intersection property. To prove \(\Leftarrow\), let \(\mathcal{A}\) be a collection of closed sets having the finite intersection property, and let \(\mathcal{B}\) be the collection of all finite intersections of elements of \(\mathcal{A}\), partially ordered by reverse inclusion.]
Solution
1. Equivalence of convergent subnets and cluster points:
1.1. A point \(x \in X\) is a cluster point of a net \((x_\alpha)_{\alpha \in A}\) if for every neighborhood \(U\) of \(x\) and every \(\alpha \in A\), there exists \(\beta \succeq \alpha\) such that \(x_\beta \in U\).
definition of net cluster point.
1.2. A net \((x_\alpha)_{\alpha \in A}\) has a convergent subnet converging to \(x\) if and only if \(x\) is a cluster point of \((x_\alpha)\).
standard characterization of subnets.
2. Forward direction (\(\implies\)): Compactness implies every net has a convergent subnet:
2.1. Let \(X\) be compact, and let \((x_\alpha)_{\alpha \in A}\) be a net in \(X\).For each \(\alpha \in A\), define the tail set \(T_\alpha = \{x_\beta \mid \beta \succeq \alpha\}\) and its closed closure \(F_\alpha = \overline{T_\alpha}\).
definition of closed tails.
2.2. The family \(\{F_\alpha\}_{\alpha \in A}\) has the Finite Intersection Property (FIP):For any finite collection \(\{\alpha_1, \dots, \alpha_n\} \subseteq A\), directedness of \(A\) provides \(\gamma \in A\) with \(\gamma \succeq \alpha_i\) for all \(1 \le i \le n\).Then \(x_\gamma \in T_{\alpha_i} \subseteq F_{\alpha_i}\) for each \(i\), so \(x_\gamma \in \bigcap_{i=1}^n F_{\alpha_i} \neq \emptyset\).
directedness of \(A\).
2.3. Since \(X\) is compact and \(\{F_\alpha\}\) has the FIP, the total intersection is non-empty:\[ \bigcap_{\alpha \in A} F_\alpha = \bigcap_{\alpha \in A} \overline{T_\alpha} \neq \emptyset. \]
FIP characterization of compactness.
2.4. Choose \(x \in \bigcap_{\alpha \in A} \overline{T_\alpha}\).For every neighborhood \(U\) of \(x\) and any \(\alpha \in A\), \(x \in \overline{T_\alpha} \implies U \cap T_\alpha \neq \emptyset\).Thus there exists \(\beta \succeq \alpha\) with \(x_\beta \in U\), so \(x\) is a cluster point of \((x_\alpha)\).By 1, \((x_\alpha)\) has a subnet converging to \(x\).
topological closure definition.
3. Reverse direction (\(\impliedby\)): Every net having a convergent subnet implies compactness:
3.1. We show that every family of closed sets with the FIP has non-empty intersection.Let \(\mathcal{A}\) be a family of closed subsets of \(X\) with the FIP.Let \(\mathcal{B}\) be the collection of all non-empty finite intersections of members of \(\mathcal{A}\):\[ \mathcal{B} = \{F_1 \cap \dots \cap F_n \mid F_i \in \mathcal{A}, \, n \ge 1\}. \]
definition of finite intersections.
3.2. Direct \(\mathcal{B}\) by reverse inclusion: \(B_1 \preceq B_2 \iff B_2 \subseteq B_1\).For any \(B_1, B_2 \in \mathcal{B}\), \(B_1 \cap B_2 \in \mathcal{B}\) with \(B_1 \cap B_2 \subseteq B_1, B_2\), so \(B_1 \cap B_2 \succeq B_1, B_2\). Thus \((\mathcal{B}, \preceq)\) is a directed set.
closure of \(\mathcal{B}\) under finite intersections.
3.3. For each \(B \in \mathcal{B}\), choose an element \(x_B \in B\).This defines a net \((x_B)_{B \in \mathcal{B}}\) in \(X\).By hypothesis, this net has a convergent subnet, hence a cluster point \(x \in X\).
Axiom of Choice and hypothesis.
3.4. For any \(F \in \mathcal{A}\), note \(F \in \mathcal{B}\).For all \(B \succeq F\) in \(\mathcal{B}\), \(B \subseteq F\), so \(x_B \in B \subseteq F\).Thus the tail \(\{x_B \mid B \succeq F\} \subseteq F\).Since \(F\) is closed and \(x\) is a cluster point of \((x_B)\), we have \(x \in \overline{\{x_B \mid B \succeq F\}} \subseteq \overline{F} = F\).
closed sets contain all cluster points of their sub-tails.
3.5. Since \(x \in F\) for every \(F \in \mathcal{A}\), we obtain \(x \in \bigcap_{F \in \mathcal{A}} F \neq \emptyset\).Thus \(X\) is compact.
FIP characterization of compactness.
4. Conclusion:\(X\) is compact if and only if every net in \(X\) has a convergent subnet. Q.E.D.
2 and 3.