Spaces where all compact subsets are closed

Question

All compact subsets of a Hausdorff space are closed and there are T$_1$ spaces (also T$_1$ sober spaces) with non-closed compact subspaces. So I looking for something in between.

Is there a characterization of the class of spaces where all compact subsets are closed? Or at least, is there a name for them?

Answer 1

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}

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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

Answer 2

According my acknowledge, it hasn't a characterization of this class of spaces, I believe that the reason is that this class of spaces is not Hausdorff and many people don't care it since we always study the classes of spaces at least Hausdorff.

The following paragraphes may be useful for you, which is copied from Page 221 Encyclopedia of General Topology:

A result taught in a first course in topology is that a compact subspace of a Hausdorff space is closed. A Hausdorff space with the property of being closed in every Hausdorff space containing it as a subspace is called H-closed (short for Hausdorff-closed).

H-closed spaces were introduced in 1924 by Alexandroff and Urysohn. They produced an example of an H-closed space that is not compact, showed that a regular H-closed space is compact, characterized a Hausdorff space as H-closed precisely when every open cover has a finite subfamily whose union is dense, and posed the question of which Hausdorff spaces can be densely embedded in an H-closed space.

Answer 3

(This is more an overlong comment:)

In a constructive reading, "all compact subsets are closed" becomes "the identity is a well-defined continuous injection from the space of compact subsets to the space of closed subsets". The latter property is equivalent to being Hausdorff (Proposition 15 in http://arxiv.org/abs/1204.3763).

Thus, for any KC-space that is not Hausdorff, proving that it is indeed KC will require some non-uniform argument.