given two topological spaces X and Y there is a proposition in Topology and Calculus which states that the smallest topology on the product topology, X x Y, such that the canonical projections: p(x,y)=x and p(x,y)=y be continous is the product topology. I need to prove this, which i cant.
I guess that one way to handle it should be the following: consider al different topologies on X x y such that the canonical projections are continous. Then one should take the intersection of all these topologies on X x Y which should be equal to the product toplogy. One should then eventually prove that this intersection ( the product topology ) is such that the canonical projections are continous. How can i show this ?
I appreciate your help. Thanks
Let $\scr{T}$ be the set of all topologies on $X \times Y$ such that $\pi_X$ and $\pi_Y$ are continuous, this set is non-empty as the discrete topology is in it.
So $\mathcal{T} = \cap \scr{T}$ is well-defined and one easily checks using the definition of a topology that $\mathcal{T}$ is a topology if all members of $\scr{T}$ are topologies (which they are).
Now if $O$ is open in $X$ resp. $Y$, then $(\pi_X)^{-1}[O]$, resp. $(\pi_Y)^{-1}[O]$ is an open set in every topology in $\scr{T}$, as these topologies are defined to make both of $\pi_X$ and $\pi_Y$ continuous. So $(\pi_X)^{-1}[O]$ (Resp. $(\pi_Y)^{-1}[O]$) is in $\mathcal{T}$ by the definition of intersection, making $\pi_X$ and $\pi_Y$ continuous w.r.t. $\mathcal{T}$, which is defined to be the product topology. This can be extended to a general existence proof for so-called initial topologies.