3 added 30 characters in body

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 Eα, where xEαy 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 is gets in one step to the same space as the ultimate relation 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?

2 Fixed typos, & made more polite.

Your construction in effect is simply 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 at a given stage, if 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 transitive closure of equivalence relation generated by the relation Eα, where xEαy 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 conforms exactly with gives rise to your construction, and 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

• Thusin fact, 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 is precisely the same as the ultimate relation 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 just because we could fix representatives of the final equivalence classes, and at each step stage before the nonHausdorff dimension, at least one point is made additional pair of points becomes equivalentto its representative.

• 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 questions 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?

1

Your construction is simply 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 at a given stage, if 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 transitive closure of the relation Eα, where xEαy 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 conforms exactly with your construction, and 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 smallest equivalence relation ∼ for which the quotient X/∼ is Hausdorff. Just prove by induction that any such relation ∼ will contain all ∼α

• Thus in fact, 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 is precisely the same relation to which your construction leads.

• The nonHausdorff dimension 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 just because we could fix representatives of the final equivalence classes, and at each step before the nonHausdorff dimension, at least one point is made equivalent to its representative.

• 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 questions 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?