Spaces where all compact subsets are closed

They seem to be usually called KC-spaces (Kompact Closed), occasionally T_B-spaces, and very rarely $J_1^\prime$-spaces. As you noticed, this class of spaces lies strictly between the T₁-spaces and the Hausdorff spaces.

I am unaware of any characterisation of them apart from the definition given. The closest thing of this kind I can think of is that a compact (not necessarily Hausdorff space) is maximally compact (i.e., no strictly finer topology is compact) iff it is KC. Additionally, any KC-space which is either first-countable or locally compact¹ is actually Hausdorff.

(As an aside, a number of questions about KC-spaces have been asked here on math.SE in the last year or so.)

1_{Here locally compact means that every neighbourhood of every point includes a compact neighbourhood of that point}

A couple references:

• A. Wilansky, Between T₁ and T₂, Amer. Math. Monthly, vol.74, no.3, pp.261-266.
• N. Smythe and C.A. Wilkins, Minimal Hausdorff and maximal compact spaces, J. Australian Math. Soc., vol.3, pp.167-171