§ Compact Hausdorff spaces are normal

Let C,DC, D be two disjoint closed subsets. We wish to exhibit disjoint opens U,VU, V which separate C,DC, D. Formally, we want CU,DV,UV=C \subseteq U, D \subseteq V, U \cap V = \emptyset. The crucial idea is to take all pairs of points in C×DC \times D, and use Hausdorffness to find opens {(Ucd,Vcd):(c,d)C×D}\{ (U_{cd}, V_{cd}) : (c, d) \in C \times D \} such that cUcd,dVcd,UcdVcd=c \in U_{cd}, d \in V_{cd}, U_{cd} \cap V_{cd} = \emptyset. which separate all pairs cc and dd, and then to use compactness to escalate this into a real separating cover. Now that we have the pairs, for a fixed c0Cc_0 \in C, consider the cover dVcd\cup_{d} V_{cd} . This covers the set DD, hence there is a finite subcover DVcDdiVcdiD \subseteq V_{cD} \equiv \cup_{d_i} V_{c {d_i}}. Now, we go back, and build the set cUcDdiUcdic \in U_{cD} \equiv \cap_{d_i} U_{c {d_i}}. This is the intersection of a finite number of opens, and is hence open. So we now have two sets UcDU_{cD} and VcDV_{cD} which separate cc from DD. We can build such a pair UcD,VcDU_{cD}, V_{cD} that separates each cc from all of DD. Then, using compactness again, we find a finite subcover of sets UciD,VciDU_{c_i D}, V_{c_i D} such that the UCDi=0nUciDU_{CD} \equiv \cup_{i=0}^n U_{c_i D} cover CC, each of the VciDV_{c_i D} cover DD (so VCDi=0nVciDV_{CD} \equiv \cap_{i=0}^n V_{c_i} D covers DD). This gives us our final opens UCDU_{CD} and VCDV_{CD}. that separate CC and DD.