Definition:Local Basis

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x$ be an element of $S$.

Also defined as
Some more modern sources suggest that in order to be a local basis, the neighborhoods of which the set $\mathcal B$ consists of do not need to be open.

Such a structure is referred to on as a neighborhood basis.

Also known as
A local basis is also known as a neighborhood basis, but that term is used on for a weaker notion.

Also see

 * Equivalence of Definitions of Local Basis
 * Definition:Local Sub-Basis
 * Local Basis is Neighborhood Basis