Enquiry on Definition of Product topology In James.E.Munkres in product topology it is written in the definition that
if X and Y are two topological spaces Then the product topology on X×Y is the topology having as basis the collection C of all sets of the form U×V, where U is an open subset of X and V is an open subset of Y. 
Now after that they have already proven  that $C $ is not a topology. Then what does the product toplogy means?? Any help would be appriciated.  thanks. 
 A: You're mixing up the definitions of topology and basis.  Munkres defines a topology as

Definition. A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ having the following properties:
$(1)$ $\emptyset$ and $X$ are in $\mathcal{T}$.
$(2)$ The union of the elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$.
$(3)$ The intersection of the elements of any finite subcollection of $\mathcal{T}$ is in $\mathcal{T}$.

In contrast, Munkres defines a basis for a topology as follows:

Definition. If $X$ is a set, a basis for a topology on $X$ is a collection $\mathfrak{B}$ of subsets of $X$ (called basis elements) 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 $B_{3}$ containing $x$ such that $B_{3} \subset B_{1}\cap B_{2}$.

Later on, Munkres proves a lemma that gives the connection between the two.

Lemma $13.1$. Let $X$ be a set; let $\mathfrak{B}$ be a basis for a topology $\mathcal{T}$ on $X$.  Then $\mathcal{T}$ equals the collection of all unions of elements on $\mathfrak{B}$.

Therefore a basis need not be a topology, but when we take all unions of elements of the basis then we obtain a topology.  In regards to the product topology, we can think of it in regards to this lemma.  Munkres defines the product topology as follows.

Definition. Let $X$ and $Y$ be topological spaces.  The product topology on $X\times Y$ is the topology having as basis the collection $\mathfrak{B}$ of all sets of the form $U\times V$, where $U$ is an open subset of $X$ and $V$ is an open subset of $Y$.

Specifically what that means is that if you take an arbitrary open set $W$  in the product topology on $X\times Y$ then it in light of the lemma we know that $$W = \bigcup_{j\in J}U_{j}\times V_{j}$$ where $J$ is some index set, $U_{j}$ is open in $X$ for each $j\in J$ and $V_{j}$ is open in $Y$ for each $j \in J.$
