In 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.

A recipe describes a cake by what was done to the ingredients. Recognising the result as a cake you already know is a different act, and it is the one that needs a rule.

A gluing instruction cuts the space into classes: each blob becomes a single point of the new space. Nothing is thrown away and nothing is added — the points of $X/\!\sim$ are precisely these blobs, and the topology is decided by which unions of them are open upstairs.

A gluing instruction generates the smallest equivalence relation containing it, and the identification space is the quotient by that relation. To recognise it, find a continuous map out of the original space onto a space already known that glues exactly what the instruction glues — and then apply the theorem below.

• The map $f$ must be continuous, onto, and must identify exactly the pairs the instruction identifies — no more and no fewer.
• $X$ must be compact and $Y$ Hausdorff. Both are needed: the conclusion comes from the fact that a continuous bijection from a compact space to a Hausdorff one is a homeomorphism.
• Collapsing a subset $A$ to a single point is written $X/A$. It is the quotient by the relation that glues all of together and leaves everything else alone.

The map is constant on classes by hypothesis, so the universal property of the quotient gives a continuous map from the glued space to the target. That map is onto, because the original map was. It is injective, because the original map glued exactly the pairs the relation glues, so two different classes have different values. So it is a continuous bijection. Its source is a quotient of a compact space and therefore compact; its target is Hausdorff. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism, which is the theorem of the separation chapter, and that finishes it.

The map is constant on classes, so it descends to the glued space.

The universal property turns continuity upstairs into continuity downstairs.

Onto because the original map was, injective because it glued exactly the classes.

A quotient of a compact space is compact, being a continuous image.

A continuous bijection from a compact space to a Hausdorff space is one.

A continuous map onto a Hausdorff space from a compact one, gluing exactly the right pairs, identifies the quotient.

• A gluing instruction generates the smallest equivalence relation containing it.
• To recognise the result, map the space continuously onto a known one, gluing exactly the same pairs.
• The source must be compact and the target Hausdorff.
• $X/A$ collapses a subset to a point.

Great work! You now know how to:

• turn a gluing instruction into an equivalence relation;
• state the recognition theorem and its two hypotheses;
• see why compactness of the source cannot be dropped;
• collapse a subset to a point.