Is empty set always part of a basis of a topology? I am reading about point-set topology.
Is empty set always part of a basis of a topology?
 A: The important property of a basis is that open sets are exactly those sets that  are unions of basis elements. 
Including the empty set in the basis, or not, does not affect which sets are unions of basis elements, so it doesn't make any difference whether the basis includes the empty set.
(Note that the empty set is always a union of basis elements, since it is the union of no sets.)
A: It can be, but it need not be: a base for a topological space $(X, \tau)$ is a family of open sets $\mathfrak{B}\subseteq\tau$ such that each $U\in\tau$ can be written as a union of elements of $\mathfrak{B}$. For instance, $\tau$ itself is a base (and $\tau$ certainly contains the empty set). However, it's also true that if $\mathfrak{B}$ is any base, then so is $\mathfrak{B}\setminus\{\emptyset\}$, that is, the set of nonempty elements of $\mathfrak{B}$ - this is because the empty set is the union of an empty collection of sets, so is a union of elements of any collection of sets.
So it can be, but it can also always be safely omitted. 
