X,Y - topological spaces, \(\displaystyle a\in X, b\in Y\)
Prove that following functions are continuous:
1.\(\displaystyle f:X\rightarrow X\times Y, f(x)=(x,b)\: \forall x \in X\)
2.\(\displaystyle g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y\)
X,Y - topological spaces, \(\displaystyle a\in X, b\in Y\)
Prove that following functions are continuous:
1.\(\displaystyle f:X\rightarrow X\times Y, f(x)=(x,b)\: \forall x \in X\)
2.\(\displaystyle g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y\)
What have you tried? It suffices to check that the preimage of basic open sets is open, right? So, let \(\displaystyle U\times V\) be basic open with \(\displaystyle U\subseteq X,V\subseteq Y\) open. Then, clearly \(\displaystyle f^{-1}(U\times V)=\begin{cases}U\quad\text{if}\quad b\in V\\ \varnothing\quad\text{if}\quad b\notin V\end{cases}\). So what?