Hausdorffization

Definition

Let $X$ be a topological space. For $a,b\in X$, define $a\sim b$ if any open set containing $a$ intersects any open set containing $b$. The Hausdorffization, also known as Hausdorffification, Hausdorffication, maximal Hausdorff quotient, or Hausdorff quotient, of $X$ is a quotient map $X\to H(X)$ with the universal property that any continuous map from $X$ to a Hausdorff space factors uniquely through the Hausdorffization.

Iterative construction

The Hausdorffization can be obtained through iterative application of the quotient map by an equivalence relation that is described below.

For a topological space $X$. First, consider the relation $\sim$ on $X$ defined as:

$a\sim b$ if for open subsets $U\ni a,V\ni b$ of $X$, $U\cap V$ is non-empty.

Note that the relation above need not itself be an equivalence relation. In particular, it need not be transitive: it is possible that $a\sim b$ and $b\sim c$ but $a\sim ̸c$.

Denote by $r_X$ the equivalence relation generated by $\sim$. Explicitly, $r_X$ is the subset of $X\times X$ that is the closure in $X\times X$ of the diagonal subspace. Denote by $h^1(X)$ the quotient space of $X$ by $r_X$, with $X\to h^1(X)$ the quotient map. In particular, this means that the points of $h^1(X)$ are the equivalence classes in $X$ under $r_X$.

Note that if $X$ is Hausdorff, then $r_X$ is a trivial relation and $h^1(X)=X$.

We now iteratively define, for every ordinal $\alpha$:

$h^{\alpha +1}(X)=h^1(h^{\alpha }(X))$

with the composite quotient maps connecting them.

For limit ordinals $\alpha$ we define $h^{\alpha }(X)$ as the direct limit of $h^{\beta }(X)$ for $\beta <\alpha$, with the quotient maps between them.

Then, $H(X)=h^{\mathrm{\infty }}(X)$.

Example to illustrate why one step isn't enough

Naively, it might seem that $h^1(X)=H(X)$, because we are already collapsing together any two points that cannot be separated by disjoint open subsets. However, this is not the case. The flaw with this chain of reasoning is that in the new quotient space, we get new open sets that can now start intersecting, even if back in the larger space, the points were separated.

Here is a concrete example of a space $X$:

• The underlying set of $X$ is the union of the set of natural numbers are $\mathrm{\infty }$.

References

• The Hausdorff Quotient by Bart Van Munster. The iterative construction here, as well as the counterexample, are described in Section 4. The counterexample is Example 4.16.