I have the topology consisting of these Sets. $$X = \{a,b,c,d,e,f\}$$ $$O = \{X,∅,\{b\},\{c,d\},\{b,c,d\},\{a,c,d,e,f\}\}$$
The question is what are the closed sets of this topology?
I have found information on the 'closure' and 'interior' of sets, thought that that was not what was asked for and then stumbled across this.
Is that what I am supposed to do?
$$X^c = \emptyset$$ $$\emptyset^c = X$$ $$\{b\}^c = \{a, c, d, e, f\}$$ $$\{c,d\}^c= \{a, b, e, f\}$$ $$\{b, c, d\}^c = \{a, e, f\}$$ $$\{a, c, d, e, f\}^c = \{b\}$$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C=\{∅,X,\{b\},\{a,e,f\},\{a,b,e,f\},\{a,c,d,e,f\}\}$$ you will obtain another set already in $C$.