A simple explanation for the product topology? Nearly every definition of the product topology is too "greek symbolish" for me to understand. So could someone explain it in simple terms, preferably how we get the $R \times R $ from two $ R's $ in the standard topology.? I mean what is the principle behind choosing sets of a certain kind in $R \times R $ versus others in being the open sets? 
 A: Let $p_1:R\times R\to R$ and $p_2:R\times R\to R$ be the projections of $R\times R\; $ to, respectively, the first  and second  co-ordinates.
Let $S$ be the set of all topologies on $R \times R$ for which $p_1$ and $p_2$ are continuous.
Let $T$ be any member of $S.$ If $U$ is an open subset of $R$ then $p_1^{-1}U=U\times R \in T.$ And if $V$ is an open subset of $R$ then  $p_2^{-1}V=R\times V \in T.$ Hence if $U,V$ are open subsets of $R$ then $U\times V=(U\times R)\cap (R\times V) \in T.$ So  the set $B=\{U\times V: U,V$ open in $R\}$ is a subset of $T.$
$\bullet\;$ $\;B$ is a base for a unique  topology $T^*$ on $R\times R.$
So $T\supset T^*$ for every $T\in S.$ Observe also that $T^*\in S$ (Because for $U,V$ open in $R$ we have $p_1^{-1}U=U\times R\in B\subset T^*$  and $p_2^{-1}V=R\times V\in B\subset  T^*.$)  
So $T^*$ is the common intersection of all the members of $S,$ and $T^*\in S.$ So $T^*$ is the weakest topology on $R\times R$ for which $p_1$ and $p_2$ are continuous. The Tychonoff product topology on $R\times R$ is defined to be be the weakest topology on $R\times R$ for which  $p_1$ and $p_2$ are continuous. So $T^*$ is the product topology.
Remark: If first sentence of the above paragraph is unclear, consider that $\forall T\in S\;(T^*\subset T)\implies T^*\subset \cap S,$ and that $T^*\in S\implies \cap S\subset T^*.$ So we have $T^*\subset \cap S\subset T^*.$
In terms of metrics, with the usual metric $d(x,y)=|x-y|$ on $R,$ we may use it to define various metrics on $R^2 .$ For example  $d_1((x,y),(x',y'))=\max (|x-x'|,y-y'|), $ and $d_2((x,y),(x',y'))=|x-x'|+|y-y'|, $ and $d_3((x,y),(x',y'))=\sqrt {(x-x')^2+(y-y')^2}.\; $  The metrics $ d_1,d_2,d_3 $ are all topologically equivalent : Each generates the product topology on $R^2.$
