A subcollection of $\mathcal B$ of a topology $T$ is a basis for $T$ if given $U\in T$ and a point $p\in U$,there exists $B\in \mathcal B$ such that $p\in B\subset U$.
But in Munkres's book, the definition of basis is different. If $X$ is a set, a basis for a topology on $X$ is a collection $B$ of subsets of $X$ such that
(1)For each $x\in X$, there is at least one basis element $B$ containing $x$.
(2)If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ such that $x\in B_3\subset B_1\cap B_2.$
Are these two definitions equivalent
Best Answer
Munkres confuses you. (and many more, judging by the many similar questions here)
In the first situation, the topology $\mathcal{T}$ is already given and you want to have a "smaller description" for it, some set of "essential" open sets, such that all open sets are unions of open sets from the base (analogous to that in a vector space all vectors are linear combinations of vectors from the basis), in topology we don't ask that this be done in a unique way, but every open set must be a union of a subcollection of the base. This is the condition expressed by
$$\forall U \in \mathcal{T}: \forall p \in U: \exists B \in \mathcal{B}: p \in B \subseteq U$$
The conditions $(1)$ and $(2)$ are for a different situation: you have a set and a collection $\mathcal{B}$ of subsets of $X$ that you want to be a base for some yet to be defined topology $\mathcal{T}$ (in the above sense). So you want the set of unions from $\mathcal{B}$:
$$\mathcal{T} = \{ \bigcup \mathcal{B}' : \mathcal{B}' \subseteq \mathcal{B}\}$$
to be a topology, and then $\mathcal{B}$ is by definition a base for $\mathcal{T}$. But it turns out that the two conditions have to be met by $\mathcal{B}$ in order to be able to prove that $\mathcal{T}$ is a topology: $(1)$ is needed to get $X \in \mathcal{T}$ and $(2)$ is for finite intersections of sets from $\mathcal{T}$ to be in $\mathcal{T}$ (recall the axioms for a topology).
So the second part solves a different problem: to define a topology by giving a base (this will be done for metric and ordered spaces, and product topologies too; it's a very basic tool). The first is to "reduce" a given topology to a more manageable subset (e.g. the intervals with rational endpoints are a countable base for $\Bbb R$) and e.g. knowing there is a countable base for a given topology is a very strong property (it sometimes lets us conclude there must be a metric on the space, e.g.).