Working with a homework problem:
$X$ and $Y$ are topological spaces and $A\subset X$, $B\subset Y$. I have $f,g:(X,A)\to (Y,B)$ as homotopic maps.
I need to show that the induced maps: $\hat{f},\hat{g}$ are homotopic.
My instinct says that the homotopy I should use is $\hat{F}:I\times (X/A)\to (Y/B)$ should be given by $\hat{F}(t,[x]) := [F(t,x)]$. I showed this map is well defined without any problem.
Edit: More precisely, $\hat{f}$ maps the pair $(t,[x])$, where $x$ is any element in $[x]\in X/A$, to $[y]$, where $y$ is any element in $[F(t,x)]$.
But I'm having difficulty showing the map is continuous. If I take $V$ to be an open subset of $Y/B$, how can I use the continuity of $F$ to show that $\hat{F}$ is continuous?