Separation Axioms

In this lesson you’ll learn the separation axioms in order, see what each one buys, understand why Hausdorff is the standard assumption, and meet the counterexamples showing the hierarchy is strict.

The three topology axioms are so weak that nothing stops a space from being useless. In the trivial topology on $\{a,b\}$, the only open sets are $\varnothing$ and the whole space, so the two points cannot be told apart by any open set, and a sequence can converge to both of them at once.

The separation axioms are increasingly strong guarantees that the topology has enough open sets to keep points and sets apart. They are numbered $T_0$ through $T_4$, and each implies the previous ones.

The figure tests three topologies on a three-point set by brute-force search over point pairs and open sets. The trivial topology fails everything, the chain topology $\{\varnothing,\{a\},\{a,b\},X\}$ satisfies $T_0$ and nothing more, and the discrete topology satisfies all three.

$T_0$. For distinct $x,y$ there is an open set containing exactly one of them. The weakest useful condition: it says the topology can at least tell points apart. Failing it means two points are genuinely interchangeable, which is why $T_0$ is assumed almost always.

$T_1$. For distinct $x,y$ there is an open set containing $x$ but not $y$, and one containing $y$ but not $x$. Equivalently:

$$T_1 \iff \text{every singleton } \{x\} \text{ is closed}$$

That reformulation is the useful one. The cofinite topology on an infinite set is $T_1$, since finite sets are exactly the closed sets.

$T_2$, Hausdorff. For distinct $x,y$ there are disjoint open sets $U\ni x$ and $V\ni y$. This is the workhorse, and three properties come with it:

• Limits are unique. Two different limits would need disjoint neighbourhoods that a convergent sequence would have to be in simultaneously.
• Compact sets are closed, as proved in lesson 10.
• The diagonal $\{(x,x)\}$ is closed in $X\times X$, which is the slick formulation.

$T_3$, regular Hausdorff. A point and a disjoint closed set can be separated by disjoint open sets. $T_4$, normal Hausdorff. Two disjoint closed sets can be. Normality is what Urysohn’s lemma needs:

In a normal space, for disjoint closed $A,B$ there is a continuous $f:X\to[0,1]$ with $f\equiv0$ on $A$ and $f\equiv1$ on $B$.

That lemma is the source of continuous functions in abstract settings, and it is why normality matters: without it you may have a topology with almost no continuous real-valued functions on it.

$$T_4\implies T_3\implies T_2\implies T_1\implies T_0$$

and each implication is strict. The counterexamples worth knowing:

Space	Satisfies	Fails
Trivial topology on $\ge2$ points	nothing	$T_0$
Sierpiński space $\{\varnothing,\{a\},X\}$	$T_0$	$T_1$
Cofinite topology on an infinite set	$T_1$	$T_2$
Various line-with-two-origins constructions	$T_2$	$T_3$
The Sorgenfrey plane	$T_3$	$T_4$

The cofinite example is the most instructive. Any two nonempty open sets there intersect, as proved in lesson 3, so no two points can be separated by disjoint sets - yet every singleton is closed. So $T_1$ and $T_2$ are genuinely different conditions.

Every metric space is $T_4$, hence $T_3$, $T_2$, $T_1$ and $T_0$.

The proof of Hausdorff is one line: for $x\ne y$ take $r=\frac{d(x,y)}{2}$ and the balls $B(x,r)$, $B(y,r)$ are disjoint by the triangle inequality.

The contrapositive is the practical use: a space failing any $T_i$ cannot be metrizable. That is how the cofinite topology on $\mathbb{Z}$ is shown to be non-metric, without constructing anything.

Full metrizability needs more, and the Urysohn metrization theorem supplies it:

A regular Hausdorff space with a countable basis is metrizable.

The two families interact, and the combinations are why “compact Hausdorff” appears so often:

• Compact + Hausdorff $\implies$ normal, so $T_4$ comes free.
• Compact Hausdorff spaces are exactly the ones where a continuous bijection is automatically a homeomorphism.
• A compact Hausdorff space is metrizable exactly when it is second countable.

Compact Hausdorff is the sweet spot of general topology: strong enough for the theorems, general enough to include all the standard examples.

Example 1: Show every metric space is Hausdorff.

Solution. Let $x\ne y$ and set $r = \frac{d(x,y)}{2}>0$.

If some $z$ lay in both $B(x,r)$ and $B(y,r)$ then

$$d(x,y)\le d(x,z)+d(z,y) < r+r = d(x,y)$$

a contradiction. So the balls are disjoint.

Answer: Hausdorff. ∎

Example 2: Show the cofinite topology on $\mathbb{Z}$ is $T_1$ but not $T_2$.

Solution. $T_1$: for $m\ne n$, the set $\mathbb{Z}\setminus\{n\}$ is open, contains $m$, and excludes $n$; symmetrically for the other direction. Equivalently, each singleton has cofinite complement, so singletons are closed ✓

Not $T_2$: any two nonempty open sets have finite complements, so the complement of their intersection is finite and the intersection is infinite, hence nonempty. No disjoint pair exists ✗

$$T_1 \text{ but not } T_2$$

∎

Consequently it is not metrizable, since every metric space is Hausdorff.

Example 3: Show limits are unique in a Hausdorff space.

Solution. Suppose $x_n\to a$ and $x_n\to b$ with $a\ne b$. Choose disjoint open $U\ni a$ and $V\ni b$.

Convergence gives $N_1$ with $x_n\in U$ for $n\ge N_1$, and $N_2$ with $x_n\in V$ for $n\ge N_2$. For $n\ge\max(N_1,N_2)$ the term $x_n$ lies in both, contradicting disjointness.

$$a = b$$

∎

In the trivial topology every sequence converges to every point, which is exactly what Hausdorff rules out.

Example 4: Show the Sierpiński space is $T_0$ but not $T_1$.

Solution. Let $X=\{a,b\}$ with $\tau=\{\varnothing,\{a\},X\}$.

$T_0$: the open set $\{a\}$ contains $a$ and not $b$ ✓

$T_1$: every open set containing $b$ is $X$, which also contains $a$. So $b$ cannot be separated from $a$ ✗. Equivalently, $\{a\}$ is not closed since its complement $\{b\}$ is not open.

$$T_0 \text{ but not } T_1$$

∎

This space is not a curiosity: it is fundamental in domain theory, where the two points represent “terminated” and “still running.”

Example 5: Show a compact Hausdorff space is normal.

Solution sketch. In a Hausdorff space, a compact set and a point outside it can be separated - the argument from lesson 10, Example 5. That gives regularity, since closed subsets of a compact space are compact.

Applying the same finite-subcover argument again, now with a compact set on each side, separates two disjoint closed sets.

Answer: normal. ∎

Both steps are the same trick: cover, take a finite subcover, intersect the finitely many neighbourhoods.

Example 6: Show $T_1$ is equivalent to all singletons being closed.

Solution. Suppose $T_1$ holds and fix $x$. For each $y\ne x$ there is an open $U_y\ni y$ with $x\notin U_y$. Then

$$X\setminus\{x\} = \bigcup_{y\ne x}U_y$$

is open, so $\{x\}$ is closed.

Conversely, if all singletons are closed then for $x\ne y$ the set $X\setminus\{y\}$ is open, contains $x$, and excludes $y$. ∎

Example 7: Use Urysohn’s lemma.

Show that in a normal space, disjoint closed sets can be separated by a continuous function.

Solution. That is exactly the statement of the lemma: there is continuous $f:X\to[0,1]$ with $f\equiv0$ on $A$ and $f\equiv1$ on $B$.

The separating open sets follow: take $U = f^{-1}\left(\left[0,\frac13\right)\right)$ and $V = f^{-1}\left(\left(\frac23,1\right]\right)$, which are open by continuity, contain $A$ and $B$, and are disjoint. ∎

Normality is what guarantees non-trivial continuous functions exist. Without it, a space can be perfectly good topologically and admit only constant maps to $\mathbb{R}$, which would make analysis on it impossible.

Domain theory and program semantics. The Sierpiński space and its relatives model observable properties of computations, where “terminates” is open and “runs forever” is not. These spaces are deliberately non-Hausdorff, and the whole theory is built on $T_0$.

Algebraic geometry. The Zariski topology is $T_1$ on a curve but not Hausdorff, so the standard analytic intuitions fail and different tools are needed. Knowing which axiom holds tells you which theorems are available.

Numerical uniqueness. A limit being unique is what lets an algorithm report “the” answer. The Hausdorff property is the abstract version of that guarantee, and non-Hausdorff quotients in optimization are a sign the model is under-specified.

Data clustering. Whether two data points can be separated by a neighbourhood is the abstract form of whether they are distinguishable by measurement. Points that cannot be separated indicate insufficient features rather than genuine identity.

Sensor resolution. A sensor grid induces a topology in which points inside the same cell are indistinguishable, so it is $T_0$ at best. Refining the grid strengthens the separation, which is exactly what increasing resolution means.

Function existence in modelling. Urysohn’s lemma is why smooth bump functions and partitions of unity exist, and those are the basic tools of differential geometry, finite element methods, and computer graphics blending.