Does the left adjoint to the forgetful functor have another left adjoint?

Question

Consider $F: Top \rightarrow Set$ the forgetful functor from the category of topological spaces into the category of sets. I know that its left adjoint is the functor which gives the set a discrete topology. Does this functor have a left adjoint of its own? Intuitively i feel like the answer should be yes, but I have no clue how to prove it.

Answer 1

The discrete topology functor $D$ does not preserve infinite products. For example, the product topology on $\prod_{n=1}^\infty D(\{ 0, 1 \})$ (which gives the categorical product in $\mathbf{Top}$) is not the discrete topology.

It follows that $D$ cannot have a left adjoint.