# Thread: Prove that following functions are continuous

1. ## Prove that following functions are continuous

X,Y - topological spaces, $a\in X, b\in Y$
Prove that following functions are continuous:
1. $f:X\rightarrow X\times Y, f(x)=(x,b)\: \forall x \in X$
2. $g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y$

2. Originally Posted by sinichko

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

P.S. This is in fact a topological embedding