T0, T1, and T2 (Hausdorff) Spaces

The separation axioms form a hierarchy of increasingly strong conditions that control how well a topology can distinguish between points and closed sets. The first three levels -- $T_0$ , $T_1$ , and $T_2$ -- concern the separation of distinct points.

The Kolmogorov Axiom ( $T_0$ )

Definition6.1$T_0$ Space

A topological space $X$ is $T_0$ (or Kolmogorov) if for every pair of distinct points $x \neq y$ , there exists an open set containing one but not the other. That is, the topology distinguishes points: $x \neq y \implies \exists U \in \tau, (x \in U, y \notin U) \text{ or } (y \in U, x \notin U)$ .

Example$T_0$ Examples

The Sierpinski space $\{0, 1\}$ with topology $\{\emptyset, \{0\}, \{0, 1\}\}$ is $T_0$ but not $T_1$ : the point $1$ has no open set separating it from $0$ .
Any $T_1$ space is $T_0$ .
The indiscrete topology on a set with more than one point is not $T_0$ .

The Fr'echet Axiom ( $T_1$ )

Definition6.2$T_1$ Space

A topological space $X$ is $T_1$ (or Fr'echet or accessible) if for every pair of distinct points $x \neq y$ , each has a neighborhood not containing the other. That is: $\forall x \neq y, \; \exists U, V \in \tau : x \in U, y \notin U, y \in V, x \notin V.$

Theorem6.1Characterizations of $T_1$

The following are equivalent for a topological space $X$ :

$X$ is $T_1$ .
Every singleton $\{x\}$ is closed.
Every finite subset of $X$ is closed.
For each $x \in X$ , $\{x\} = \bigcap\{U : U \text{ open}, x \in U\}$ .

Proof

(1 $\Rightarrow$ 2): For $y \neq x$ , there exists an open set $U_y$ with $y \in U_y$ and $x \notin U_y$ . Then $X \setminus \{x\} = \bigcup_{y \neq x} U_y$ is open, so $\{x\}$ is closed.

(2 $\Rightarrow$ 3): Finite unions of closed sets are closed.

(3 $\Rightarrow$ 1): Given $x \neq y$ , the set $X \setminus \{y\}$ is open (since $\{y\}$ is closed) and contains $x$ but not $y$ . Similarly, $X \setminus \{x\}$ is open and contains $y$ but not $x$ .

(2 $\Leftrightarrow$ 4): $\{x\} = \bigcap\{U : x \in U\}$ iff for every $y \neq x$ there is an open set containing $x$ but not $y$ , which together with the analogue for the other point gives the result (since singletons being closed is equivalent).

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The Hausdorff Axiom ( $T_2$ )

Definition6.3$T_2$ (Hausdorff) Space

A topological space $X$ is $T_2$ (or Hausdorff) if for every pair of distinct points $x \neq y$ , there exist disjoint open sets $U, V$ with $x \in U$ and $y \in V$ : $\forall x \neq y, \; \exists U, V \in \tau : x \in U, y \in V, U \cap V = \emptyset.$

RemarkThe Hausdorff Condition is Very Common

Most spaces encountered in analysis and geometry are Hausdorff:

All metric spaces are Hausdorff ( $U = B(x, d/2)$ , $V = B(y, d/2)$ where $d = d(x, y) > 0$ ).
All manifolds are Hausdorff (by definition in most conventions).
All subspaces and products of Hausdorff spaces are Hausdorff.
Quotient spaces of Hausdorff spaces need not be Hausdorff.

ExampleHausdorff and Non-Hausdorff Examples

Hausdorff: $\mathbb{R}^n$ , $S^n$ , all metric spaces, all manifolds, $\mathbb{R}$ with the lower limit topology.

$T_1$ but not Hausdorff: The cofinite topology on an infinite set. Singletons are closed (so $T_1$ ), but any two nonempty open sets intersect (their complements are finite, so they cannot be disjoint).

Not $T_1$ : The Sierpinski space.

Consequences of the Hausdorff Property

Theorem6.2Properties of Hausdorff Spaces

Let $X$ be a Hausdorff space.

Limits of sequences are unique: If $x_n \to x$ and $x_n \to y$ , then $x = y$ .
The diagonal $\Delta = \{(x, x)\}$ is closed in $X \times X$ .
Compact subsets are closed (Theorem 5.16).
Continuous maps into Hausdorff spaces are determined by their values on a dense subset: if $f, g: X \to Y$ are continuous, $Y$ is Hausdorff, and $f|_D = g|_D$ for a dense $D \subseteq X$ , then $f = g$ .

Proof

(1): If $x \neq y$ , choose disjoint open $U \ni x$ and $V \ni y$ . Then $x_n \in U$ for large $n$ and $x_n \in V$ for large $n$ , contradicting $U \cap V = \emptyset$ .

(2): Let $(x, y) \notin \Delta$ , i.e., $x \neq y$ . Choose disjoint open $U \ni x$ , $V \ni y$ . Then $U \times V$ is an open neighborhood of $(x, y)$ in $X \times X$ , and $(U \times V) \cap \Delta = \emptyset$ . So $X \times X \setminus \Delta$ is open.

(4): Let $E = \{x : f(x) = g(x)\}$ . Then $E = (f, g)^{-1}(\Delta)$ where $(f, g): X \to Y \times Y$ is continuous. Since $\Delta$ is closed in $Y \times Y$ (by (2)), $E$ is closed. Since $D \subseteq E$ and $D$ is dense, $E = X$ .

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RemarkThe Hierarchy So Far

$T_2 \implies T_1 \implies T_0$ Each implication is strict. The separation axioms continue with regularity ( $T_3$ ) and normality ( $T_4$ ), which involve separating points from closed sets and closed sets from each other.

T0, T1, and T2 (Hausdorff) Spaces

The Kolmogorov Axiom (T0T_0T0​)

The Fr'echet Axiom (T1T_1T1​)

The Hausdorff Axiom (T2T_2T2​)

Consequences of the Hausdorff Property

The Kolmogorov Axiom ( $T_0$ )

The Fr'echet Axiom ( $T_1$ )

The Hausdorff Axiom ( $T_2$ )