Separation axioms measure how strongly a topology can distinguish points and closed sets. They are the hypotheses that make limits unique, compact sets closed, and function-extension theorems possible.
The point-separation hierarchy
The first separation axioms concern points alone.
Definition. A space is \(T_0\) if any two distinct points are topologically distinguishable by an open set containing one but not the other. It is \(T_1\) if each point is closed. It is Hausdorff, or \(T_2\), if distinct points have disjoint open neighborhoods.
Proposition. Hausdorff implies \(T_1\), and \(T_1\) implies \(T_0\).
Proof. If \(X\) is Hausdorff and \(x\ne y\), choose disjoint open neighborhoods \(U\ni x\) and \(V\ni y\). For every \(y\ne x\), there is an open set containing \(y\) but not \(x\), so \(X\setminus\{x\}\) is open and \(\{x\}\) is closed. Thus \(X\) is \(T_1\). If points are closed, then \(X\setminus\{y\}\) is an open set containing \(x\) but not \(y\), so \(T_0\) holds.
Example. The Sierpinski space \(\{0,1\}\) with open sets \(\varnothing,\{1\},\{0,1\}\) is \(T_0\) but not \(T_1\).
Regular and normal spaces
Stronger axioms separate points from closed sets, or closed sets from closed sets.
Definition. A \(T_1\) space is regular if for every closed \(F\) and point \(x\notin F\), there are disjoint open sets \(U,V\) with \(x\in U\) and \(F\subseteq V\). A \(T_1\) space is normal if disjoint closed sets can be separated by disjoint open neighborhoods.
Proposition. Normal implies regular, and regular implies Hausdorff.
Proof. If \(X\) is normal and \(F\) is closed with \(x\notin F\), then \(\{x\}\) and \(F\) are disjoint closed sets because \(T_1\) makes singletons closed; normality separates them. If \(X\) is regular and \(x\ne y\), apply regularity to the closed set \(\{y\}\) and the point \(x\).
Compactness improves separation
Compactness can convert pointwise Hausdorff separation into global separation from compact sets.
Proposition. In a Hausdorff space, every compact subset is closed.
Proof. Let \(K\subseteq X\) be compact and \(x\notin K\). For each \(y\in K\), choose disjoint open sets \(U_y\ni y\) and \(V_y\ni x\). The sets \(U_y\) cover \(K\); choose \(U_{y_1},\dots,U_{y_n}\) covering \(K\). Then \(V=\bigcap_{k=1}^nV_{y_k}\) is an open neighborhood of \(x\) disjoint from \(K\). Hence \(X\setminus K\) is open.
Corollary. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
Proof. If \(F\subseteq X\) is closed, then \(F\) is compact. Its image under the continuous map is compact, hence closed in the Hausdorff target. Thus the bijection is a closed map, so its inverse is continuous.
Function separation
Normality is powerful because it creates continuous real-valued functions separating closed sets.
Theorem. Urysohn's lemma says that if \(X\) is normal and \(A,B\subseteq X\) are disjoint closed sets, then there is a continuous \(f:X\to[0,1]\) with \(f(A)=0\) and \(f(B)=1\).
Proof. The detailed dyadic construction is given in the Urysohn lemma page. The separation axiom supplies nested open sets whose closures are properly contained in later open sets; the dyadic labels on these sets define \(f\), and normality ensures continuity.
Remark. This is the bridge from pure open-set topology to analysis on topological spaces.