Answer to Question #141464 in Differential Geometry | Topology for sabah sazan

Question

Q/Let T be the class of subsets of N consisting of empty set and all subscts of the form G_n ={n,n +1,n+ 2,..} with n € N. Show that T is topology on N.

Answer 1

$G_1=N$ clearly. So the entire and empty set belongs to $T.$ Now we note $G_n\supseteq G_{n+1}.$ Hence $\cup_{} G_{k}$ is equal to $G_{i}$ with $i$ the minimum index in the collection. Hence the union is in $T.$ Now for finite intersection, given any finite collection there is a maximum say $i.$ Then $\cap G_k=G_i$ here $i$ is the maximum in the collection. Hence finite intersection and arbitrary union is there. Hence $T$ is a topology.