Recent content by son

this is what i came up with... but this is not consider a proof... every element in C will be in the basis β_C. Let U be in C then U is the finite intersection of elements in C, for example U = U ∩ U. It follows that U ∈ β_C. And by the definition of the topology, every element in β_C is...

let X be a set and C be a collection of subsets of X whose union equal X. let β[SUB]C the collection of all subsets of X that can be expressed as an intersection of finitely many of the sets from C. let T[SUB]C be the topology generated by the basis β[SUB]C. prove that every set in C is...

Let F={F_i} be a collection of closed sets. Then F_i=X-U_i for some collection {U_i} of open sets of X, because of the definition of closed. Then De-Morgans rules give intersection F_i = intersection (X-U_i) = X - (union U_i) But union U_i is an open set because the unions of open sets are...

the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open. but I am not sure how i would start the proof...

the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open. but I am not sure how i would start the proof...

the theorem i am using for a closed set is... Let X be a topological space. the following statements about the collection of closed set in X hold: (i) the empty set and X are closed (ii) the intersection of any collection of closed sets is a closed set (iii) the union of finitely many closed...

Prove that the intersection of any collection of closed sets in a topological space X is closed. Homework Statement Homework Equations The Attempt at a Solution