Prove every set in C is open in the topology generated by basis β

[son]

let X be a set and C be a collection of subsets of X whose union equal X. let βC the collection of all subsets of X that can be expressed as an intersection of finitely many of the sets from C.

let TC be the topology generated by the basis βC.

prove that every set in C is an open set in the topology TC.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[lanedance]

any ideas/attempts?

 

[sutupidmath]

Essentially what you need to do is show that any subset A of C can be expressed as a union of elements of B_C.

C in this case is a sub-basis of X, and T_c the topology that this subbasis induces, so to speak.

 

[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 open, so U is thus open.