Topology · Lesson 08

Chapter practice

Twelve 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.

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

x \neq = y

T_{2}

T_{1}

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

Chapter navigation

Intuition

Twelve 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.

A building code is a list of demands, and the interesting buildings are the ones that meet some and fail others. Knowing which is which is the whole of the subject here.

Before you practise

The chapter in five lines.

T_2\implies T_1\implies T_0,\qquad T_4\implies T_3\implies T_2

What to remember

$T_0$ tells two points apart one way, $T_1$ both ways, and $T_1$ is the same as every point being closed.
Hausdorff gives disjoint open sets around two points; metric spaces have it and the cofinite topology does not.
In a Hausdorff space limits are unique and compact subsets are closed.
Regular separates a point from a closed set; normal separates two closed sets. With $T_1$ they become $T_3$ and $T_4$ .
Second countable means a countable basis; first countable means one at each point; the Sorgenfrey line separates the two.