Separable and firstcountable not implies secondcountable
This article gives the statement and possibly, proof, of a nonimplication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
View a complete list of topological space property nonimplications  View a complete list of topological space property implications Get help on looking up topological space property implications/nonimplications

Contents
Statement
Verbal statement
It is possible for a topological space to be both separable and firstcountable, but not secondcountable.
Definitions used
Separable space
Further information: Separable space
A topological space is termed separable if it has a countable dense subset.
Firstcountable space
Further information: Firstcountable space
A topological space is termed firstcountable if for every point in the space, there is a countable basis at that point.
Secondcountable space
Further information: Secondcountable space
A topological space is termed secondcountable if it admits a countable basis.
Converse
Any secondcountable space is both firstcountable and separable. Further information: Secondcountable implies firstcountable, secondcountable implies separable
Examples
Sorgenfrey line
The Sorgenfrey line, which is defined as the real numbers given the lower limit topology for the usual ordering, is firstcountable and separable but not secondcountable.
For full proof, refer: Sorgenfrey line is firstcountable, Sorgenfrey line is separable, Sorgenfrey line is not secondcountable