/sci/ - Science & Math

Bumping out of abstract interest.

>>11588514
First of all, if we have an open set [math]X[/math] which contains [math]A[/math], then it's preimage at any copy [math]I_n[/math] contains a half-open interval up to some real number, that is, [math]\pi ^{-1} (X) \cap I_n \supseteq [0, a)_n[/math] for some [math]a>0[/math], by the definition of quotient topology, right?
So for whatever countable basis, you just need to construct an element which isn't in any basis element by making him stop halfway before this [math]a[/math] at some given coordinate.
So essentially, we have a real-valued function [math]f(X)[/math] such that [math]\pi ^{-1} (X_n) \cap I_n \supseteq [0, f (X_n) ~ )[/math]. Then if we have a supposed countable basis given by the [math]A_n[/math], we can construct [math]L \subset \prod I_n[/math] given by [math]L = \prod _n [0, f (A_n)/2 )[/math] whose projection thus doesn't contain any element of the supposed base.

My bad for any typos, possible mistakes and for the confusing notation, the products and the quotients also confuse me.