- #1
demonelite123
- 219
- 0
i want to prove that given a set K in G that is closed and a set F in K that is closed, then K is closed in G. K, F, and G are all topological spaces.
so to reword the problem i instead want to show that given G-K is open in G and K-F is open in K show that G-F is open in G. so since G is a topological space and K is a subspace of G, i endow K with the subspace topology. then there exists some open set O in G such that any open subset of K such as K-F = O ∩ K. then F = K - (O∩K) then i drew myself a picture and found that F also equals (G-O)∩K which works because G-F = G-(G-O) ∪ (G-K) = O U (G-K) and since O is open in G and (G-K) is open in G therefore the union of those 2 sets is also open in G and G-F is then open in G which means F is closed in G.
i wish to know that in these types of proofs, is it common to have to draw pictures in order to continue? i was stuck at F = K - (O∩K) for a while until i drew a picture and saw that F = (G-O)∩K which i could take the complement of since it was a subset of G. my question is could F = (G-O)∩K be derived from F = K - (O∩K) or any of the other information i was given, without having to draw pictures to derive it? or is drawing pictures the usual way of going through these set theory types of proofs. thanks.
so to reword the problem i instead want to show that given G-K is open in G and K-F is open in K show that G-F is open in G. so since G is a topological space and K is a subspace of G, i endow K with the subspace topology. then there exists some open set O in G such that any open subset of K such as K-F = O ∩ K. then F = K - (O∩K) then i drew myself a picture and found that F also equals (G-O)∩K which works because G-F = G-(G-O) ∪ (G-K) = O U (G-K) and since O is open in G and (G-K) is open in G therefore the union of those 2 sets is also open in G and G-F is then open in G which means F is closed in G.
i wish to know that in these types of proofs, is it common to have to draw pictures in order to continue? i was stuck at F = K - (O∩K) for a while until i drew a picture and saw that F = (G-O)∩K which i could take the complement of since it was a subset of G. my question is could F = (G-O)∩K be derived from F = K - (O∩K) or any of the other information i was given, without having to draw pictures to derive it? or is drawing pictures the usual way of going through these set theory types of proofs. thanks.