Monomorphisms in spaces with Lindel\"of filters via some compact-open-like topologies on C(X)

We study the monomorphisms in the category LSpFi of spaces with Lindel"{o}f Filters. We extend the criterion derived by R. Ball and A. Hager in “Monomorphisms in Spaces with Lindel"{o}f Filters”, Czech. Math.J. 57, No.1 (2007) 281-317. Based on this extension, we defined and studied some compact-open-like topologies on C(X) and give new characterizations of the monomorphisms in LSpFi via these topologies.