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Thread: Prove that following functions are continuous

  1. #1
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    Prove that following functions are continuous

    Help me please

    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.$\displaystyle g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y$
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sinichko View Post
    Help me please

    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?


    P.S. This is in fact a topological embedding
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