Topology · Chapter 08

Separation and countability

How much a topology can tell apart, and how little of it needs writing down: the separation ladder from T0 to normal, what Hausdorff buys, and the two countability axioms the familiar spaces satisfy.

U_{x} \cap U_{y} = \emptyset

x \neq = y

T_{2}

T_{1}

B = {B_{n} : n \in N}

B_{x}

{x} = {x}

F \cap G = \emptyset

Lessons

Build the foundation

01Telling points apartEverything so far has been stated in open sets, and open sets can be very coarse: in the indiscrete topology no open set distinguishes one point from another, so the space cannot see its own points. The separation axioms are a ladder of demands, each asking the open sets to tell more apart. The first two rungs ask about points only, and the second of them turns out to be the same thing as every single point being closed.Read lesson →
02Hausdorff spacesThe next rung asks for room: two distinct points should have open sets around them that do not meet at all. This is the Hausdorff condition, and it is the one almost every space in analysis and geometry satisfies. It is strictly stronger than the rung below, and the cofinite topology is what shows that.Read lesson →
03Why Hausdorff mattersA condition earns its name by what it lets you prove. The Hausdorff property is what makes a limit unique: without it a sequence can settle on two points at once, and "the limit" is not a phrase that means anything. In the cofinite topology every sequence of distinct points converges to every point of the space at the same time, which is exactly how bad it gets.Read lesson →
04Compact sets in Hausdorff spacesThe compactness chapter left a gap: a compact subset need not be closed, and the Sierpiński space showed it. The missing hypothesis is the Hausdorff property, and with it the implication holds in every space, not only in metric ones. The proof is the same shape as the metric one, with disjoint open sets in place of balls of half the distance.Read lesson →

05Regular spacesThe ladder continues by replacing one of the two points with a closed set. A space is regular when a point outside a closed set can be separated from the whole of it by disjoint open sets. The useful form of the condition is a shrinking statement: inside any open set around a point there is a smaller open set whose closure still fits.Read lesson →

06Normal spacesThe last rung replaces both points with closed sets: two disjoint closed sets should have disjoint open sets around them. This is normality, and it is the condition under which continuous functions can be built where none were given — the content of Urysohn's lemma, which this course names and leaves to a later one. Normality is also the rung that behaves worst: unlike the others it is not inherited by subspaces.Read lesson →

07First and second countabilityA topology can be enormous — the open sets of the line are already uncountably many — and yet be described by a countable list. Second countability asks for a countable basis for the whole space, first countability only for a countable basis at each point. Both hold in the spaces this course is built on, and the second is what makes sequences enough to detect closures and continuity.Read lesson →

08Chapter practiceTwelve questions across the chapter, with no teaching blocks. Every rung of the ladder is a demand of the open sets, and every counterexample in this chapter is a space where the open sets are too large or too few to meet it.Read lesson →

Mapping the lesson.

Separation and countability

Build the foundation