# Prove that following functions are continuous

#### sinichko

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.$$\displaystyle g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y$$

#### Drexel28

MHF Hall of Honor
X,Y - topological spaces, $$\displaystyle a\in X, b\in Y$$
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?