Topological Space in which every compact subset is metrizable

Solution 1:

There is, but the area of math that it was studied in is basically dead and multiple terms were used to describe these and related spaces. Your spaces are called KM-spaces, derived from the (perhaps tongue-in-cheek?) more thoroughly studied KFC-spaces, which were sometimes instead called Kelley Spaces, or $k$-spaces, or $k'$-spaces. So it's a mess. I would not expect the typical topologist to be familiar with the term.

F. Siwiec, KM-spaces and km-covering mappings, Not. Amer. Math. Soc. 18 (1971), 1109-1110 #71T-G200.

https://www.jstor.org/stable/2037385

https://www.jstor.org/stable/44236412 (historical exposition)