- #1

radou

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## Homework Statement

So, if X has a countable basis {Bn}, then every basis C for X contains a countable basis for X.

## The Attempt at a Solution

First of all, consider all the intersections of elements of C of the form Ci[tex]\cap[/tex]Cj. For every x in the intersection (if it's non empty), choose a basis element Bx contained in the intersection. There are at most countable such elements. Then, for every Bx, choose an element Cx of C contained in Bx, and containing x. There are at most countable such elements, too. Now do this for all the possible intersections Ci[tex]\cap[/tex]Cj.

Since X has a countable basis, every open covering for X contains a countable subcollection covering X. Apply this to C in order to obtain a countable subcollection C'. We need to show that C' is a basis for X.

First, C' obviously covers X, so there's only one more condition left - we need to show that if x is in the intersection Ci[tex]\cap[/tex]Cj of elements of C', then there exists an element Ck from C' such that x[tex]\in[/tex]Ck[tex]\subseteq[/tex]Ci[tex]\cap[/tex]Cj.

The part I'm not sure about is this:

*if C' is countable, then the number of intersections of two elements of C' is countable.*If this is true, then for each intersection, take the countable number of sets Cx from the first paragraph, and add it to C'. A countable union of countable sets is countable, hence the new collection C'' is a countable subcollection of C which is a basis for X.