I lately looked up the definition of being a compactly generated space on Wikipedia:
Definition: A topological space $X$ is compactly generated if it satisfies the following condition: A subspace $A$ is closed in $X$ if and only if $A \cap K$ is closed in $K$ for all compact subspaces $K \subset X$.
In the second sentence it says: "Equivalently, one can replace closed with open in this definition". I tried to show that this is actually equivalent, but I did not succeed. Bascially you have:
$A$ open $\iff$ $(X\backslash A)$ closed $\iff \forall K $ compact: $K \cap (X\backslash A)$ closed
What one would need is pulling the set difference out, namely:
$\forall K $ compact: $K \cap (X\backslash A)$ closed $\iff \forall K $ compact: $X\backslash(K \cap A) $ closed $\iff \forall K $ compact: $K \cap A$ open.
But is the first equivalence true? I could not find a justification for it, probably I am not seeing a simple argument here.