Product of two topologies? Is the Cartesian product of two topologies again topology?
According to my knowledge , NO ? So how the product of two or many topologies defined ?
Could someone make it clear for me. I am new to topology.
 A: A topology on $X$ is a set of subsets of $X$. If $\tau$ is a topology on $X$ and $\sigma$ a topology on $Y$, then $\tau\times\sigma$ is not a subset of $X\times Y$, in general, because it's a set of pairs of subsets of $X$ and $Y$.
What you can consider is $\mathcal{B}=\{U\times V:U\in\tau, V\in\sigma\}$, that's not the same as $\tau\times\sigma$, but is a set of subsets of $X\times Y$.
However, $\mathcal{B}$ is generally not a topology, because it lacks the property of being closed under unions. Since
$$
(U_1\times V_1)\cap(U_2\times V_2)=
(U_1\cap U_2)\times(V_1\cap V_2)
$$
we deduce $\mathcal{B}$ is closed under finite intersections. In particular $\mathcal{B}$ is a basis for the topology $\hat{\mathcal{B}}$ consisting of arbitrary unions of members of $\mathcal{B}$.
This is the product topology, that is, the least topology on $X\times Y$ making the projection maps $X\times Y\to X$ and $X\times Y\to Y$ continuous.
It's clear that every set of the form $U\times V$ with $U\in\tau$ and $V\in\sigma$ must be open in any topology making those maps continuous. Since the least topology containing $\mathcal{B}$ is $\hat{\mathcal{B}}$, we are done.
A: This question does not make any sense. You have to define the topology on the product $X\times Y$ by urself. However there are many useful/common topologies for the product of the topological spaces. The most common one is the product-topology. Another example is the box-topology, which is totally the same as the product-topology for finite products.
Since you are new to topology. Topology is about defining the open sets by yourself. In your analysis course you considered different function on a fixed topological space (mostly $\mathbb R$ with the standard topology). New it's vice versa. You may look at a fixed function with different topologies. 
A question could be: What is the smallest (in terms of inclusion) topology such that a given function is continuous?.. But this would take us to a different topic.
