nonhausdorff dimension

Question

if $X$ is a topological space, a first step in making $X$ hausdorff is taking the quotient $H(X)=X/\sim$, where $\sim$ is the equivalence relation generated by: if $x,y$ cannot be seperated by disjoint open sets, then $x \sim y$. observe that $X$ is hausdorff, when $X \to H(X)$ is an isomorphism, and that for every hausdorff space $K$ the map $Hom(H(X),K) \to Hom(X,K)$ induced by the projection $X \to H(X)$ is bijective.

by a fairly general categorical argument, we can construct from this the free functor from topological spaces to hausdorff spaces (i.e. it's left adjoint to the forgetful functor): for ordinal numbers $\alpha$, define the functor $H^\alpha$ (together with natural transformations $H^{\alpha} \to H^{\beta}, \alpha < \beta$) by $H^0 = id, H^{\alpha+1} = H \circ H^\alpha$ and $H^\alpha = colim_{\delta < \alpha} H^\delta$. for every topological space $X$ there is an ordinal number $\alpha$ such that $H^\alpha(X) = H^{\alpha+1}(X)$, then $H^\alpha(X)$ is the free hausdorff space associated to $X$. define $h(X)$, the "nonhausdorff dimension" to be the smallest such ordinal number $\alpha$. every ordinal number arises as a nonhausdorff dimension(!).

I've came up with this with a friend and we don't know of any literature about it. perhaps someone of you has already seen it elsewhere? there are some further questions: every $H^\alpha(X)$ is a quotient of $X$, but how can we describe the equivalence relation explicitely? what is the intuition for a space $X$ to have nonhausdorff dimension $\alpha$? are there known classes of topological spaces whose nonhausdorff dimension can be bounded? and of course: is there some use for the nonhausdorff dimension? ;-)

Answer 1

Your construction in effect is defining an increasing transfinite sequence of equivalence relations on the space X, as Mariano correctly describes in his comment. The point is to identify points whenever they would be a violation of the Hausdorff property in the quotient by the relation constructed so far.

One can implement this idea uniformly (without splitting into separate successor and limit cases) as follows. For any topological space X and any ordinal α, let ∼_α be the equivalence relation generated by the relation E_α, where xE_αy if and only if whenever U and V are open sets containing x and y, then there is some x' in U and y' in V and some β<α such that x'∼_βy'.

This relation gives rise to your spaces; the α-th space is simply the quotient X/∼_α. It is easy to see from this definition that ∼₀ is just =, that successor stages do what you want, that β<α implies that ∼_β subset ∼_α, and that ∼_λ is the union of the earlier ∼_β for limit ordinals λ.

If the quotient X/∼_α is Hausdorff, then the relation stops growing, since no more identifications are made, and the least α for which this occurs is what you called the nonHausdorff dimension.

Some easy observations:

• The resulting Hausdorff space X/∼_α at the dimension α is the obtained from smallest equivalence relation ∼ for which the quotient X/∼ is Hausdorff. Just prove by induction that any such relation ∼ will contain all ∼_α

• Thus, there is also a top-down description of the resulting Hausdorff space: Let ∼ be the intersection of all equivalence relations E on X, whose quotient x/E is Hausdorff. This gets in one step to the same space as the ultimate Hausdorff space to which your construction leads.

• The nonHausdorff dimension of an infinite space is an ordinal whose cardinality is no larger than the cardinality of the original space. (For example, the dimension of a countable space will be a countable ordinal.) This is because at each stage before the nonHausdorff dimension, at least one additional pair of points becomes equivalent.

• The dimension of the disjoint union of many spaces will be the supremum of their individual dimensions.

• In other words, if a space is disconnected, written as the disjoint union of open sets, then inductively the equivalence relations will never cross between these sets. Thus, the dimension of the whole space will be the supremum of the dimensions of these open subspaces.

I have a question of my own about this construction. Namely, let us call the first quotient space X¹ the nonHausdorff derivative of X, borrowing terminology from the case of Cantor-Bendixon. My question is: is every space the nonHausdorff derivative of another space? In other words, is there a nonHausdorff anti-derivative?

Answer 2

What you have defined is what general topologists call an ordinal invariant of a topological space. This is just what it sounds like: an assignment of an ordinal number to every topological space in such a way that homeomorphic spaces get assigned equal ordinals.

Knowing this terminology may help you search the literature to see whether your invariant already appears. For instance, you might look here:

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Kannan, V. Ordinal invariants in topology. Mem. Amer. Math. Soc. 32 (1981), no. 245, v+164 pp.

MathReview by S.P. Franklin:

What follows is the text of an advertising blurb for this manuscript used by the AMS. Filtering out the obvious sales pitch leaves a general but accurate account of the contents: ``The concept of the order of a map is so powerful as to form a base for the unification of several ordinal invariants in topology. In this work, the author shows that the derived length of scattered spaces, sequential order of sequential spaces, etc., can all be described in terms of this notion. This view helps to extend them so as to be defined for all topological spaces without missing their most significant properties, to dualize them, to perceive them in the background of category theory and to obtain a lot of new information. In this self-contained work the author incidentally comes across some close connections among such apparently unrelated areas of topology as Čech closures, coreflective subcategories, special morphisms and the ordinal invariants mentioned above. The notion of $E$-order introduced here provides a unification of such invariants as sequential order, $k$-order, $m$-net-order and so on. This theory is not only more satisfactory than the earlier attempts of unification but also encompasses them as subcases.''

The list beginning with sequential order should also include the derived order.

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To me it seems possible that this definition and your result -- that every ordinal number arises in this way from a topological space -- could be publishable, if written up in a succinct and attractive way.

Answer 3

An unpublished, but quite well distributed, reference is "Appendix A, Compactly Generated Spaces" to the 1978 Ph.D. thesis of L. Gaunce Lewis, Jr. Proposition 4.1 discusses the existence of the left adjoint (Hausdorffification) by way of Freyd's adjoint functor theorem, and Construction 4.3 explains the transfinite induction by iterating $X \to JX$ where JX is essentially the quotient of X discussed by Martin Brandenburg. (Lewis works with weak Hausdorff spaces, but the story is the same.) A published version of something much like this appears in Lewis, May and Steinberger's book "Equivariant Stable Homotopy Theory" (Springer LNM 1213), where the spectrification functor L is constructed in Section 1 of the Appendix (pages 475-481) by this kind of transfinite induction.

Answer 4

I have been interested in a related problem for a long time. For a C*-algebra $A$, the space Prim($A$) of primitive ideals of $A$ with the hull-kernel topology is seldom Hausdorff and one considers its complete regularization Glimm$(A)$. For primitive ideals $P$ and $Q$ one writes $P\sim Q$ if $P$ and $Q$ cannot be separated by disjoint open sets in Prim$(A)$. With this relation, Prim$(A)$ becomes a graph and then Orc$(A)$ is the supremum of the diameters of connected components of this graph. The case when Orc$(A)$ is finite is reasonably well understood (at least when Prim$(A)$ is compact) but little is known about what can happen when Orc$(A)$ is infinite, which would correspond more with the discussion above.

I suppose that my question at this stage is whether the work on non-Hausdorff dimension has now been published; and also in what situations would the Hausdorffization coincide with the complete regularization?