What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ? 
Given two spaces $X$ and $Y$, how to define the mapping space betweeen them, i.e. what topology should we put on the set of maps between them? 

If $X$ is compact and $Y$ a metric space, this is quite easy as one can put a metric on $Map(X,Y)$: For $f,g\in Map(X,Y)$ define their distance just to be the maximum of the distances between $f(x)$ and $g(x)$ as $x$ ranging over the points in $X$. 

If $Y$ is no longer metric, we have to find a replacement what it means for two maps to be close. Say, we have again two maps $f,g\in Map(X,Y)$. Let $K\subset X$ be compact and $U\subset Y$ be open such that $f(K)\subset U$. Assume now that $Y$ is Hausdorff (else, this construction might behave badly anyhow). Then $f(K)\subset Y$ is closed, so you would expect that if you move $f(K)$ a little bit, then it stays inside of $U$. So if $g$ is close to $f$, then $g(K)$ should be still inside of $U$. 
Thus, it is sensible to define an open neighborhood of $f$ to be all maps $g$ such that $g(K) \subset U$. And actually, this agrees with the metric definition when $X$ is compact and $Y$ metric! Furthermore, as long as you have "enough" compact subsets in $X$ (i.e. $X$ is compactly generated) and everything is Hausdorff, then this topology has all the pleasant properties one likes. 
