Separation and connectedness · Definition

Hausdorff space

topologyseparationtopological spaces

Before you start you need to know

It is the minimal condition for the points of a space to be told apart by neighbourhoods, and with it limits stop being ambiguous.

Definition

A topological space \(X\) is Hausdorff if for every pair of distinct points \(x\) and \(y\) there exist disjoint open sets \(U\ni x\) and \(V\ni y\).

\[x\neq y\ \Rightarrow\ \exists\,U,V\ \text{open}:\ x\in U,\ y\in V,\ U\cap V=\varnothing\]

The idea is that two distinct points have open neighbourhoods that do not touch. In \(\mathbb{R}\), and in every metric space, this holds: if \(r=d(x,y)>0\), the balls of radius \(r/2\) around \(x\) and \(y\) are disjoint by the triangle inequality. The property is also called the separation axiom \(T_2\).

Finite spaces show how strong it is: a finite Hausdorff space is discrete, because each point is separated from the others and the finite intersection of those open sets gives \(\{x\}\). In the trivial topology, or the cofinite topology on an infinite set, any two nonempty open sets meet and no separation is possible.

The property is inherited: every subspace of a Hausdorff space is Hausdorff, and so is a product of Hausdorff spaces. Moreover, in a Hausdorff space every point is a closed set and, if \(f:X\to Y\) is continuous and injective with \(Y\) Hausdorff, then \(X\) is too. Closed singletons are necessary but not sufficient: the cofinite topology on \(\mathbb{N}\) has the first and not the second.

Its practical value is that in a Hausdorff space limits of sequences are unique and compact sets are closed. Quotients, on the other hand, can destroy it, so it has to be checked when gluing or identifying points.

Think about it

Why is every metric space Hausdorff?

If \(x\neq y\) and \(r=d(x,y)\), the open balls of radius \(r/2\) around each are disjoint: a common point \(z\) would give \(d(x,y)\le d(x,z)+d(z,y)<r\).

Is \(\mathbb{R}\) with the cofinite topology Hausdorff?

No: two nonempty open sets have finite complements, so their intersection is infinite and never empty.

Exercises 10

  1. Which of these topologies on \(X=\{1,2,3\}\) is Hausdorff?
  2. How many of the 29 possible topologies on a 3-point set are Hausdorff?
  3. Which of these topologies on \(\mathbb{R}\) is not Hausdorff?
  4. In \(X=\{1,2,3,4\}\) with open sets \(\varnothing, \{1\}, \{2\}, \{1,2\}, \{3,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}\), how many pairs of distinct points can be separated by disjoint open sets?
  5. If \(X\) is Hausdorff and \(A\subseteq X\), the subspace \(A\) is:
  6. In a Hausdorff space \(X\), what can be said about each singleton \(\{x\}\)?
  7. In \(X=\{1,2,3,4\}\) with open sets \(\varnothing, \{1\}, \{4\}, \{1,4\}, \{2,3\}, \{1,2,3\}, \{2,3,4\}, \{1,2,3,4\}\), which pair of points cannot be separated by disjoint open sets?
  8. Let \(f:X\to Y\) be continuous and injective, with \(Y\) Hausdorff. What follows about \(X\)?
  9. Let \(X=\{a,b\}\) have the trivial topology and \(Y\) any Hausdorff space with two or more points. Is \(Y\times X\) with the product topology Hausdorff?
  10. Let \(\mathbb{N}\) have the cofinite topology. Which statement is correct?

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §17
  • Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 3