Topology · Chapter 09

Quotients and constructions

Building spaces by gluing: the topology a quotient carries, the rule that recognises the result, and the circle, cylinder, torus, projective plane and one-point compactification it builds.

q : X \to X / \sim

[x]

S^{1} \times S^{1}

0 \sim 1

RP^{n}

X^{+} = X \cup {\infty}

q^{- 1} (U)

S^{1}

Lessons

Build the foundation

01The quotient topologyEvery construction so far has taken spaces apart: subspaces, products, components. Gluing goes the other way. Decide which points are to become the same, take the set of groups, and there is exactly one sensible topology on it: a set of groups is open when the points it is made of form an open set below. That is the quotient topology, and it is the finest topology making the gluing map continuous.Read lesson →
02Identification spacesIn practice a gluing is not given as an equivalence relation but as an instruction on a picture: join these two edges, collapse this circle to a point, identify opposite points of the rim. Each instruction generates a relation, and the quotient is the space it builds. The problem is then always the same: the quotient is described by how it was made, and the answer wanted is a space already known. One theorem settles that, and this chapter uses it four times.Read lesson →
03Building the circle from an intervalThe first construction is the one the whole chapter is modelled on: take a closed interval and glue its two ends together. What comes out is the circle, and the proof is the recognition theorem applied to the map that wraps the interval once round.Read lesson →
04Building a cylinderGluing one pair of opposite edges of a square gives a cylinder. The picture is a sheet rolled up, and the proof is the same as for the circle with an extra coordinate carried along untouched. Gluing the same pair with a flip gives a different space, the Möbius band, which is the first sign that how the edges are joined matters as much as which.Read lesson →

05

Building a torusGlue both pairs of opposite edges of the square and the result is a torus, the surface of a doughnut. It is also the product of two circles, which says something worth keeping: a space built by gluing can turn out to be a space built by multiplying, and the two descriptions are equally good.

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06Real projective spaceA line through the origin is decided by any non-zero point on it, and two points decide the same line exactly when one is a multiple of the other. Taking the set of lines through the origin as a space is the projective construction, and it is a gluing: on the sphere, identify each point with the point opposite it. In one dimension the result is a circle again; in two it is a surface that does not fit inside ordinary space.Read lesson →

07One-point compactificationA space that is not compact can often be made compact by adding a single point. The new point is thought of as infinity, and a set around it is declared open when what it leaves out is compact. The line becomes a circle, the plane becomes a sphere, and the construction is the standard way of putting a boundary on a space that had none.Read lesson →

08Course reviewFourteen questions across the whole course, with no teaching blocks. The course began with a distance and ended with spaces built by gluing, and every step in between was made in open sets alone. These questions are drawn from all nine chapters.Read lesson →

Mapping the lesson.

Quotients and constructions

Build the foundation