Prove that following functions are continuous

Apr 2010
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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.\(\displaystyle g:Y\rightarrow X\times Y, g(y)=(a,y)\: \forall y \in Y\)
 

Drexel28

MHF Hall of Honor
Nov 2009
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Berkeley, California
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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