Topology · Lesson 06

Chapter practice

Twelve questions across the chapter, with no teaching blocks. A basis, a subbasis and a subspace all describe a topology by something smaller than itself, and deciding which of the three a question is about is part of what is being practised.

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τ_{Y} = {U \cap Y : U \in τ_{X}}

B

U = α ⋃ B_{α}

B_{1} \cap B_{2}

Y \subseteq X

Chapter navigation

Intuition

Twelve questions across the chapter, with no teaching blocks. A basis, a subbasis and a subspace all describe a topology by something smaller than itself, and deciding which of the three a question is about is part of what is being practised.

A mechanic is handed an engine, not a list of symptoms. Which tool the job needs is the first decision and often the whole of it.

Before you practise

The chapter in five lines.

\mathcal{B}\rightsquigarrow\tau:\ \text{unions};\qquad \mathcal{S}\rightsquigarrow\mathcal{B}:\ \text{finite intersections}

What to remember

A basis is a family of open sets whose unions are the open sets; the point test is: $x\in B\subseteq U$ .
A family generates a topology when it covers the set and fits inside its own overlaps around each point.
A subbasis needs only to cover; its finite intersections are a basis, and it generates the coarsest topology making its members open.
The subspace topology on $A$ is the traces $U\cap A$ , and a set open in $A$ need not be open in $X$ .
Finer means more open sets, and it is checked through bases one point at a time.