nLab main separation axioms -- as lifting properties

In terms of lifting properties

The separation conditions $T_0$ to $T_4$ may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others.

This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.

In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:

• points (elements) are denoted by $\bullet$ with subscripts indicating where the points map to;

• boxes are put around open subsets,

• an arrow $\bullet_u \to \bullet_c$ means that $\bullet_c$ is in the topological closure of $\bullet_u$.

In the lifting diagrams for $T_2-T_4$ below, an arrow out of the given topological space $X$ is a map that determines (classifies) a decomposition of $X$ into a union of subsets with properties indicated by the picture of the finite space.

Notice that the diagrams for $T_2$-$T_4$ below do not in themselves imply $T_1$.

Proposition

(Lifting property encoding $T_0$)
The following lifting property in Top equivalently encodes the separation axiom $T_0$:

Proposition

(Lifting property encoding $T_1$)
The following lifting property in Top equivalently encodes the separation axiom $T_1$:

Proposition

(Lifting property encoding $T_2$)
The following lifting property in Top equivalently encodes the separation axiom $T_2$:

Proposition

(Lifting property encoding $T_3$)
The following lifting property in Top equivalently encodes the separation axiom $T_3$:

Proposition

(Lifting property encoding $T_4$)
The following lifting property in Top equivalently encodes the separation axiom $T_4$: