# Preimage of a set in the product topology

Claim: When $$X ×Y$$ is endowed with the product topology $$T_{X×Y}$$ , the projection maps $$p_X : X × Y → X , p_X(x, y) = x$$, and $$p_Y : X × Y → Y , p_Y (x, y) = y$$ , are continuous.

Proof: Indeed for any open set $$U$$ in $$X$$ , $$p^{−1}_X (U) = U × Y$$

This is a claim from my lecture notes. However, the projection map is clearly not injective. How can we define the preimage of a non-injective map? Clearly, $$p_X(x,y_1)=x=p_X(x, y_2)$$ for any $$y_1\neq y_2$$

For any function $$f$$ an any set $$A$$ in the range the notation $$f^{-1}(A)$$ stands for all points $$x$$ in the domain of $$f$$ such that $$f(x)\in A$$. If $$f$$ happens to have an inverse then $$f^{-1}(A)$$ becomes the image of $$A$$ under $$f^{-1}$$.
• I see. So we are defining a map $f^{-1}:X \rightarrow X$ x $Y$ by $f^{-1}(t)=t$ x $Y$ rather than taking the inverse of $f$, is that correct? Will mark the answer as correct when the website allows me to! – asdf Dec 5 '18 at 9:42