According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:
A topological space $X$ is exponentiable iff $X$ is core-compact.
The category of core-compact topological spaces is not cartesian closed, because even though when $X$ and $Y$ are core-compact, we can build the exponential object $Y^X$, but $Y^X$ will not necessarily be itself core-compact.
The category of Core-compactly generated spaces is cartesian closed.
The category of Core-compactly generated spaces contains the category of core-compact spaces.
It seems that there is a contradiction if the four propositions are all true. A sub-category of the category of topological spaces cannot be cartesian closed if it is not strictly included in the sub-category of core-compact spaces. What Am i getting wrong here ?