Munkres Introduction to Topology: Section 17 Problem 1

The collection $C$ fits the exact criteria for a collection of closed sets, so it follows that the compliment, $X \ - \ C_n$ for $C_n \ \in \ C$ is an open set and that the compliments of all $C_n \ \in \ C$ forms a collection of closed sets. We can show that this collection (which we will call $T_c$) will form a topology by simply checking the criteria:

1. We are given that for $T_c \ = \ \{X \ - \ C_n \ | \ C_n \ \in \ C\} \ \Rightarrow \ \emptyset, \ X \ \in \ C$. This means that $X \ - \ \emptyset \ = \ X$ and $X \ - \ X \ = \ \emptyset$ are in $T_c$.



2. It is stated that arbitrary intersections of $C_n$ elements are in $C$. This means that $\bigcap_n \ C_n \ \in \ C$. We can write a general element of $T_c$ in this form as $X \ - \ \bigcap_n \ C_n$. By De Morgan's Law, we get:
   
   $$X \ - \ \bigcap_n \ C_n \ = \ \bigcup_n \ X \ - \ C_n$$
   
   This shows that arbitrary unions of elements of $T_c$ are in $T_c$: the second criteria for $T_c$ to be a topology.



3. By the same logic as criteria $2 \ \Rightarrow \ \bigcup_n \ C_n \ \in \ C$ for finite $n$. A general element of $T_c$ is written as $X \ - \ \bigcup_n \ C_n$ for finite $n$. This means that by De Morgan's Law:
   
   $$X \ - \ \bigcup_n \ C_n \ = \ \bigcap_n \ X \ - \ C_n$$
   
   This hold true for finite $n$. This is the third criteria for $T_c$ being a topology, by definition.