Seemingly Contradicting Basis Concepts in Munkres's Book and that of Willard In S. Willard's "General Topology" he states the following theorem:

However in J. Munkres' "Topology" the author states the following lemma:

It seems to me that they are contradictory, because if for instance we consider discrete topology on some set $X$, then it is evident that $2^{X}=\mathcal{T}\neq X$. Can anyone please elucidate whether I am wrong in my deduction or not and if I am, where is my error?
 A: The Willard fact is about a set $X$ that has no topology yet. Then to define a (unique) topology on $X$ you can use a base, i.e. a family $\mathcal B$ with the two stated properties (which are just internal to $\mathcal B$ and do not refer to a topology yet) and then this newly defined topology is just the set of all unions from $\mathcal B$, i.e.
$$\mathcal T = \{O\mid \exists \mathcal B' \subseteq \mathcal B: O = \bigcup \mathcal B'\}\tag{1}$$
The fact in Munkres comes from another angle: he also discusses the properties that Willard does (they're necessary and sufficient and quite natural) but he starts with a set with topology $(X,\mathcal T)$ and a base for that topology (not a base to define a new topology) is a family $\mathcal B \subseteq \mathcal T$ (so basic sets are open in particular) so that $$\forall O \in \mathcal T: \forall x \in O: \exists B_x \in \mathcal B: x \in B_x \subseteq O\tag{2}$$
It turns out (and that is what Munkres shows in his Lemma) that a base for $\mathcal T$ as he defines it in $(2)$ is actually also a base that we could have used to define the topology as in $(1)$ (!).
So the books are perfectly consistent but take a different angle of approach, as it were.
For the discrete topology $\mathcal T=2^X$ we can use the base $\mathcal B=\{\{x\}\mid x \in X\}$ which trivially obeys the two conditions of Willard; $(b)$ is void, essentially) and as any set $A$ is a union of its singletons i.e. $A = \bigcup \{\{x\}\mid x \in A\}$ we see that every subset of $X$ is indeed a union of base sets and all those unions form the discrete topology, i.e. the power set. And $(2)$ is also clear for this base. But we don't need a base to define the discrete topology as $2^X$ obviously fulfills the 3 topology axioms anyway, but just to illustrate..
