1) If $\mathcal{A}$ consists of the Sierpinski space and $\mathcal{B}$ consists of all finite topological spaces then the concrete coreflective hull of $\mathcal{A}$ consists of all finitely generted spaces (i.e. spaces in which $x \in clA \iff x \in cl\{a\}$ for some $a \in A$). This can be seen by showing that the total sink from Sierpinski spaces is final in Top.
So let $X$ be a (finitely generated) topological space and take the total sink from Sierpinski spaces (the functions in this sink are named $f_{i}$). Let $g: X \rightarrow Y$ be a function, so that for every $f_{i}: Sierpinski \rightarrow X$, $g \circ f_{i}$ is continuous. Why is g continuous?
2) If $\mathcal{A}$ consists of one convergent sequence $A= \{0\} \cup \{\frac{1}{n} \vert n \in \mathbb{N} \} $ or if $\mathcal{B}$ consists of the full subconstruct of all metrizable topological spaces, then in both cases the concretely coreflective hull consists of all sequential spaces. Again this is seen by showing that the total sink from $A = \{0\} \cup \{\frac{1}{n} \vert n \in \mathbb{N}\}$ is final in Top.
So let $X$ be a (sequential) topological space and take the total sink from A (the functions in this sink are named $f_{i}$). Let $g: X \rightarrow Y$ be a function, so that for every $f_{i}: A \rightarrow X$, $g \circ f_{i}$ is continuous. Why is g continuous?
3) If Sierpinski spaces belong to $\mathcal{A}$, then the concretely reflective hull of $\mathcal{A}$ in Top is Top. To prove this show that for an arbitrary topological space, the source consisting of all indicatorfunctions defined on open sets is initial. Again, I'm stuck proving this argument.
As always, any help would be appreciated.