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Bases and subspaces · Topology | Math Infinitum

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Topology
Bases and subspaces

Topology · Chapter 03

Bases and subspaces

Describing a topology by a small family that generates it, and the topology a subset inherits from the space around it.

Describing a topology by a small family that generates it, and the topology a subset inherits from the space around it.

τ_{Y} = {U \cap Y : U \in τ_{X}}

B

U = α ⋃ B_{α}

B_{1} \cap B_{2}

Y \subseteq X

τ_{1} \subseteq τ_{2}

⟨ S ⟩

B_{x}

Lessons

Build the foundation

01Bases for a topologyListing every open set of the line is hopeless; describing them all as unions of intervals takes one sentence. A family that does this for a topology is called a basis, and most topologies are met through one rather than through the collection itself.Read lesson →
02Generating a topology from a basisThe last lesson started with a topology and found a family inside it. This one runs the other way: start with a family of subsets and ask whether unions of them form a topology. Two conditions are enough, and meeting them is how most topologies are actually defined.Read lesson →
03SubbasesA basis has to fit inside every overlap. Drop that demand and ask only that the family cover the space: take all the finite intersections first, and a basis appears. Such a family is called a subbasis, and it is how a topology is described when only a few conditions matter.Read lesson →
04The subspace topologyAny subset of a space becomes a space in its own right. Its open sets are the traces the surrounding open sets leave on it, and nothing else has to be invented.Read lesson →
05Comparing topologiesOne set can carry many topologies, and they are compared by containment: more open sets means finer. Doing this by listing is hopeless for the line, so the comparison is made through bases — and the test is the point test again, run between two families.Read lesson →

06Chapter practiceTwelve 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.Read lesson →