I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$):
A weak basis of a topological space $X$ is a collection of nonempty open sets s.t. every nonempty open set contains one of them.
My question is: is this definition equivalent to that of a basis for a topology?
The fact that the author gives a specific name to such a family suggests that it is not, but for every $x\in X$ and for every open nhbd $U(x)$ there exists $V(x)$ in the weak basis contained in $U$. This means that a weak basis is also a covering and hence satisfies the conditions for being a basis.
Any comment is appreciated. Thank you in advance for your help.