Quotients, Products and Identifications | Building New Topological Spaces

Topology does not only analyse spaces. It gives precise machines for building new ones.

Two of the most important construction machines are products and quotients. A product places spaces side by side so that a point records several coordinates at once. A quotient starts with a space and deliberately declares selected points equivalent, collapsing or gluing them into single points. Together these constructions create familiar objects such as planes, cylinders, circles and tori, and they prepare the language for covering spaces, manifolds, algebraic topology and geometric topology.

This guide serves R22.04 · Quotients, products and identifications in the BTT Mathematics Atlas. It continues the route through Topological Spaces and Continuity, Metric Spaces and Examples, and Compactness and Connectedness.

Return route: BTT Mathematics Hub.

The simple answer

Given topological spaces X and Y, the product topology on X×Y is the topology generated by sets U×V with U open in X and V open in Y. It is the natural topology that makes the coordinate projections to X and Y continuous.

Given a surjection q:X→Y, the topology on Y is the quotient topology induced by q when a subset U⊆Y is declared open exactly when q⁻¹(U) is open in X. In the common identification setting, q sends each point to its equivalence class. The quotient space treats every equivalence class as a single point.

Products separate coordinates. Quotients identify points. Both constructions are designed around continuity.

1. Cartesian products before topology

For sets X and Y, the Cartesian product is X×Y={(x,y):x∈X,y∈Y}. A point in the product contains one coordinate from each factor.

The projection maps are

• π_X:X×Y→X, π_X(x,y)=x;
• π_Y:X×Y→Y, π_Y(x,y)=y.

Before choosing a topology, these are only set maps. The product topology is chosen so that they become continuous in the least complicated natural way.

2. The product topology on two factors

Let X and Y be topological spaces. The sets U×V, with U open in X and V open in Y, form a basis for a topology on X×Y. This is the product topology.

A general open set in X×Y need not itself be a single rectangle U×V. It is a union of such basic rectangles.

Worked example 1: R²

Take R with its usual topology. The product topology on R×R=R² has basis elements (a,b)×(c,d), ordinary open rectangles. This topology agrees with the topology induced by the Euclidean metric on R².

A Euclidean open disc is not usually a single basic rectangle, but every point of the disc lies inside a sufficiently small open rectangle contained in the disc. Conversely, every open rectangle is Euclidean-open. Thus the two descriptions generate the same open sets.

3. Why the projections are continuous

Let U be open in X. Then

π_X⁻¹(U)=U×Y,

which is open in the product topology because Y is open in itself. Therefore π_X is continuous. The same argument proves π_Y is continuous.

In fact, the product topology is the coarsest topology on X×Y making both coordinate projections continuous. Any topology with continuous projections must contain all sets U×Y and X×V, and therefore all finite intersections U×V and their unions.

4. Continuity into a product is coordinatewise

Suppose f:Z→X×Y is given by f(z)=(f₁(z),f₂(z)). Then f is continuous if and only if both coordinate functions f₁:Z→X and f₂:Z→Y are continuous.

If f is continuous, then f₁=π_X∘f and f₂=π_Y∘f are compositions of continuous maps. Conversely, if both coordinates are continuous, then for a basic open rectangle U×V,

f⁻¹(U×V)=f₁⁻¹(U)∩f₂⁻¹(V),

which is open. Therefore f is continuous.

Worked example 2: a parametrised curve

Define γ:R→R² by γ(t)=(cos t,sin t). Both coordinate functions are continuous, so γ is continuous into the product R×R. Its image is the unit circle S¹.

This coordinatewise criterion is used constantly in geometry and analysis.

5. Maps out of a product are different

Continuity into a product can be checked coordinate by coordinate. Continuity out of a product has no analogous rule saying that separate continuity in each variable automatically gives joint continuity.

A function F:X×Y→Z may be continuous in x for each fixed y and continuous in y for each fixed x while failing to be jointly continuous. For example, define F:R²→R by F(0,0)=0 and, away from (0,0),

F(x,y)=xy/(x²+y²).

For fixed y=0 or fixed x=0 the value is 0, and each one-variable slice is continuous. But along y=x≠0, F(x,x)=1/2, so F does not tend to 0 as (x,y)→(0,0). Joint topology detects simultaneous variation of coordinates.

6. Finite products and many coordinates

For finitely many spaces X₁,…,Xₙ, a basis for the product topology consists of sets U₁×⋯×Uₙ with each Uᵢ open in Xᵢ.

For an arbitrary family ∏_i∈IXᵢ, the product topology is generated by sets restricting only finitely many coordinates at a time: basic sets have the form ∏Uᵢ where Uᵢ is open in Xᵢ and Uᵢ=Xᵢ for all but finitely many indices.

This finite-coordinate rule is important. If arbitrary coordinates were simultaneously restricted, the result would be the generally finer box topology, not the product topology. For finite products the two notions coincide; for infinite products they may differ dramatically.

7. Products preserve familiar topological properties

Several major properties behave well under products:

• A product of connected spaces is connected.
• A product of path-connected spaces is path connected.
• A finite product of compact spaces is compact; more generally, arbitrary products of compact spaces are compact in the product topology by Tychonoff’s theorem.
• A product of Hausdorff spaces is Hausdorff.

These preservation laws allow large spaces to inherit structure from smaller factors.

8. Cylinders and tori as products

The cylinder can be represented as S¹×[0,1]. A point records an angular position around the circle and a height. Because S¹ and [0,1] are compact and connected, their product is compact and connected.

The torus can be represented as S¹×S¹. Each point records two circular coordinates. This product description is mathematically cleaner than relying only on the familiar doughnut picture; the same topology can be studied without choosing a particular embedding in R³.

9. Equivalence relations and identification spaces

A relation ∼ on X is an equivalence relation if it is reflexive, symmetric and transitive. The equivalence class of x is

[x]={y∈X:y∼x}.

The quotient set X/∼ is the set of all equivalence classes. The canonical projection q:X→X/∼ sends x to [x].

As a set, X/∼ only records which points have been identified. The quotient topology determines how neighbourhoods behave after the identification.

10. The quotient topology

Let q:X→Y be a surjection. The quotient topology on Y induced by q is defined by

U⊆Y is open if and only if q⁻¹(U) is open in X.

With this topology, q is automatically continuous. More strongly, it is a quotient map: openness downstairs is exactly detected by taking inverse images upstairs.

A continuous surjection need not be a quotient map. A quotient map is a continuous surjection with the additional property that U is open in the codomain whenever q⁻¹(U) is open in the domain.

11. Worked example: identifying the endpoints of an interval

Start with [0,1] and declare 0∼1, while every other point is equivalent only to itself. The quotient [0,1]/(0∼1) is homeomorphic to the circle S¹.

Define f:[0,1]→S¹ by

f(t)=(cos2πt,sin2πt).

The map is continuous and surjective, and f(0)=f(1). Its fibres are exactly the equivalence classes: the endpoints share a fibre and interior points have single-point fibres. Therefore f factors through a bijection f̄:[0,1]/∼→S¹.

The interval is compact and S¹ is Hausdorff. The induced continuous bijection f̄ from the compact quotient to S¹ is a homeomorphism. Topologically, gluing the two endpoints closes the interval into a loop.

12. Saturated subsets

A subset A⊆X is saturated with respect to q if it is a union of complete fibres of q; equivalently, A=q⁻¹(q(A)).

If points in a fibre have been identified, a saturated set either contains the whole fibre or none of it. Saturated open sets are especially useful because if q is an open quotient map, their images are open. Even without openness of q, quotient topology can always be tested by pulling sets back.

13. The universal property of a quotient

The most useful theorem about quotient spaces is a factorisation principle.

Let q:X→X/∼ be the quotient map. Suppose f:X→Z is continuous and constant on equivalence classes: x∼y implies f(x)=f(y). Then there is a unique set map f̄:X/∼→Z satisfying f=f̄∘q. Moreover, f̄ is continuous.

The quotient therefore captures exactly the information that remains after the identified points are declared indistinguishable for the map f.

Proof of continuity

For an open set V⊆Z,

q⁻¹(f̄⁻¹(V))=f⁻¹(V),

which is open because f is continuous. By the defining property of the quotient topology, f̄⁻¹(V) is open in X/∼. Thus f̄ is continuous.

14. A square becomes a cylinder

Take the square [0,1]×[0,1]. Identify (0,y) with (1,y) for each y∈[0,1], while making no other new identifications. The left and right edges are glued point-for-point. The quotient is homeomorphic to S¹×[0,1], a cylinder.

The product description and quotient description are two views of the same topological space. One says “circle coordinate plus height”; the other says “rectangle with one pair of opposite edges glued”.

15. A square becomes a torus

Again start with [0,1]². First identify the left and right edges in the same orientation: (0,y)∼(1,y). Then identify the bottom and top edges in the same orientation: (x,0)∼(x,1).

The resulting quotient is a torus T², homeomorphic to S¹×S¹.

The four corners all become one point because the equivalence relation is transitive through the edge identifications. This illustrates why one must define the full equivalence relation, not merely draw arrows on a diagram and stop there.

16. Twisted identifications: Möbius strip

Start with the rectangle [0,1]×[−1,1]. Identify the vertical edges with a reversal:

(0,t)∼(1,−t).

The quotient is a Möbius strip. Unlike the ordinary cylinder, the gluing reverses the transverse coordinate. This change has a global consequence: the Möbius strip is non-orientable and has one boundary component rather than two.

The phrase “twist the strip” is useful geometric intuition, but the equivalence relation is the precise topological instruction.

17. Other edge identifications

Different patterns of edge identification produce different spaces.

• Same-orientation gluing of both pairs of square edges produces a torus.
• One pair with a reversal and the other with a compatible gluing can produce a Klein bottle.
• An appropriate boundary identification of a polygon can produce the real projective plane.

These are not merely different drawings of one object. The resulting spaces can have different invariants such as orientability and fundamental group. Later topology develops tools that prove they are not homeomorphic.

18. Collapsing a subspace to a point

If A⊆X, define an equivalence relation by making all points of A equivalent to each other and leaving points outside A equivalent only to themselves. The quotient is written X/A. It collapses A to a single point.

For example, collapsing the boundary Sⁿ⁻¹ of the closed n-ball Dⁿ to one point produces a quotient homeomorphic to the sphere Sⁿ. In the case n=1, collapsing both boundary points of an interval together gives S¹. In the case n=2, collapsing the boundary circle of a disc to one point gives S².

This construction is fundamental in algebraic topology because it turns a subspace into a distinguished basepoint and produces spheres and cell attachments naturally.

19. Quotient maps are not simply open maps

A quotient map need not be open, and it need not be closed. What defines it is the inverse-image criterion for open subsets of the codomain.

However, every surjective open continuous map is a quotient map, and every surjective closed continuous map is a quotient map. These are useful sufficient conditions.

Proof for the open case: suppose q is continuous, surjective and open, and q⁻¹(U) is open. Because q is surjective, U=q(q⁻¹(U)). Since q is open, U is open. Thus q satisfies the quotient criterion.

20. A continuous surjection that is not quotient

It is useful to know that continuity plus surjectivity is not enough. Let X be R with the discrete topology and Y be R with the usual topology, and let q:X→Y be the identity set map. Since every subset of X is open, q is continuous: the inverse image of every usual open set is open in X. It is also bijective, hence surjective.

But it is not a quotient map. If it were, every subset U of Y would be open because q⁻¹(U)=U is open in discrete X. That would force Y to be discrete, contradicting its usual topology.

This example makes the quotient condition concrete.

21. Quotients preserve compactness and connectedness

A quotient map q:X→Y is continuous and surjective. Therefore:

• If X is compact, then Y=q(X) is compact.
• If X is connected, then Y=q(X) is connected.
• If X is path connected, then Y is path connected.

The last claim follows because paths can be composed with q. Quotienting can identify points and simplify a space, but it cannot destroy these properties through a continuous surjection in the forward direction.

22. Hausdorffness can fail under quotients

Unlike compactness and connectedness, Hausdorffness is not automatically preserved by quotient maps. Poorly behaved identifications can make distinct equivalence classes impossible to separate by disjoint neighbourhoods.

A useful theorem in compact Hausdorff settings is that if X is compact Hausdorff and the equivalence relation is sufficiently closed—technically, if {(x,y)∈X×X:x∼y} is closed in X×X—then the quotient X/∼ is Hausdorff. The extra hypothesis controls how equivalence classes accumulate.

This is a good example of a general topological habit: preservation theorems need their hypotheses attached.

23. Subspaces, products and quotients are different operations

• A subspace keeps some points and inherits open sets by intersection.
• A product combines coordinates from several spaces.
• A quotient identifies selected points or fibres.

These operations can be combined, but they solve different construction problems. Confusing them leads to incorrect topology.

24. Products and subspaces do not always commute naively

If A⊆X and B⊆Y carry their subspace topologies, then A×B with the product topology agrees with the subspace topology inherited from X×Y. This useful compatibility follows because a basic open set in A×B is (A∩U)×(B∩V)=(A×B)∩(U×V).

But more elaborate operations—especially quotients combined with products—may require additional hypotheses to behave as a simple diagram suggests. Topology rewards checking the universal property rather than relying only on pictures.

25. Product of quotient maps: a caution

If q:X→Y is a quotient map, it is tempting to assume q×id_Z:X×Z→Y×Z is always a quotient map. This is not true for arbitrary spaces Z. It does hold under useful conditions, for example when Z is locally compact Hausdorff.

This is a more advanced warning, but it illustrates an important principle: natural-looking constructions can fail at the boundary of a theorem’s hypotheses. When products and quotients interact, use an established criterion rather than assuming formal compatibility.

26. Identifications and topological invariants

Edge-gluing pictures are useful for proposing spaces, but topology needs invariants to distinguish the results. Compactness and connectedness are two such invariants, though often too coarse. Later cells introduce the fundamental group, covering spaces, homology and cohomology.

For instance, a circle and a closed interval are both compact and connected, but they are not homeomorphic. Removing an interior point disconnects the interval into two pieces, while removing one point from a circle leaves a connected space homeomorphic to an open interval. That deletion behaviour is already a topological distinction.

27. A construction workflow

When a product or quotient appears, use a disciplined sequence.

1. State the underlying sets and topologies.
2. For a product, identify the factor topologies and a basis of product opens.
3. For a quotient, state the equivalence relation or surjection precisely.
4. Write the projection or quotient map explicitly.
5. Use inverse images to prove continuity.
6. Use the product or quotient universal property when factoring maps.
7. Check compactness, connectedness, Hausdorffness and other invariants under the exact hypotheses available.
8. Only then use a geometric name such as circle, cylinder or torus after a homeomorphism has been justified.

This workflow prevents a common error: treating a suggestive diagram as if it were already a proof of topological equivalence.

28. Common misconceptions

• “Every open set in a product is a rectangle.” False. Rectangles form a basis; general opens are unions of them.
• “The product topology on infinitely many factors lets every coordinate be restricted at once.” False. Basic product opens restrict only finitely many coordinates.
• “Separate continuity implies joint continuity.” False in general.
• “Any surjective continuous map is a quotient map.” False.
• “A quotient map must be open.” False.
• “Gluing is just a picture.” The equivalence relation and quotient topology are the precise construction.
• “Quotients preserve Hausdorffness.” Not automatically.
• “A cylinder and Möbius strip come from the same edge gluing.” Their identifications differ by a reversal and have different global topology.
• “All corners remain distinct after square-edge gluing.” Transitivity of the equivalence relation can identify them.

29. Worked proof: the diagonal is closed in a Hausdorff space

Let X be Hausdorff and consider the diagonal Δ={(x,x):x∈X}⊆X×X. We show Δ is closed.

Take (x,y)∉Δ, so x≠y. By Hausdorffness there are disjoint open sets U containing x and V containing y. Then U×V is an open product neighbourhood of (x,y) disjoint from Δ. Thus every point outside Δ has an open neighbourhood outside Δ. The complement of Δ is open, so Δ is closed.

This proof is a compact illustration of how product topology translates separation properties into geometric subsets of X×X.

30. Worked proof: product paths

Suppose X and Y are path connected. To join (x₀,y₀) to (x₁,y₁), choose paths α:[0,1]→X from x₀ to x₁ and β:[0,1]→Y from y₀ to y₁. Define γ(t)=(α(t),β(t)). The coordinate functions are continuous, so γ is continuous by the product criterion. Hence X×Y is path connected.

The proof uses the universal behaviour of products rather than a special metric formula.

31. Practice set

1. Describe a basis for the product topology on R×R.
2. Prove each coordinate projection X×Y→X and X×Y→Y is continuous.
3. Show f:Z→X×Y is continuous iff both coordinate functions are continuous.
4. Explain why an open disc in R² is open in the product topology.
5. Prove A×B has the same topology whether formed as a product of subspaces or as a subspace of X×Y.
6. Show S¹×[0,1] is compact and connected.
7. Show S¹×S¹ is path connected.
8. Define precisely the equivalence relation on [0,1] that identifies the endpoints.
9. Show t↦(cos2πt,sin2πt) is constant exactly on the required endpoint equivalence classes.
10. State the quotient universal property.
11. Prove a surjective open continuous map is quotient.
12. Give a continuous surjection that is not quotient.
13. Explain why a quotient of a compact space is compact.
14. Explain why a quotient of a connected space is connected.
15. Describe the edge identification that turns a square into a cylinder.
16. Describe the edge identifications that turn a square into a torus.
17. Describe the reversal that produces a Möbius strip from a rectangle.
18. What does it mean for a subset to be saturated?
19. Show that collapsing the boundary of D¹=[−1,1] to one point produces a circle.
20. Explain why product topology and box topology differ for infinitely many factors.

32. Answers and checks

1. Open rectangles U×V with U,V open in R.
2. π_X⁻¹(U)=U×Y and similarly for π_Y.
3. Use composition with projections in one direction and inverse images of basis rectangles in the other.
4. Every point of the disc has a sufficiently small open rectangle contained in the disc.
5. Basic opens are (A∩U)×(B∩V)=(A×B)∩(U×V).
6. Both factors are compact and connected; finite products preserve both properties.
7. Both circle factors are path connected; coordinatewise product paths join any two product points.
8. x∼y iff x=y, or {x,y}={0,1}.
9. The map has f(0)=f(1), and on [0,1) the angle 2πt gives distinct points of S¹.
10. A continuous map constant on equivalence classes factors uniquely through a continuous map from the quotient.
11. If q⁻¹(U) is open, then U=q(q⁻¹(U)) is open because q is surjective and open.
12. The identity from discrete R to usual R is continuous and bijective but not quotient.
13. A quotient map is continuous and surjective; continuous images of compact spaces are compact.
14. Continuous images of connected spaces are connected.
15. Identify (0,y) with (1,y) for every y, with no reversal.
16. Identify opposite vertical edges in the same orientation and opposite horizontal edges in the same orientation.
17. Identify (0,t) with (1,−t).
18. It is a union of complete fibres: A=q⁻¹(q(A)).
19. The two boundary points are identified; parameterising the interval around S¹ gives the homeomorphism.
20. Product-basic sets restrict only finitely many coordinates, while box-basic sets may restrict every coordinate.

33. What to carry forward

Products and quotients are not merely vocabulary. They are universal construction principles. Product topology is the natural setting for several coordinates and makes maps into products coordinatewise continuous. Quotient topology is the natural setting for identification and makes maps out of a quotient correspond to maps on the original space that respect equivalence classes.

Once spaces can be built by gluing, the next question is how to detect holes and how local pieces cover a global space. That leads naturally to fundamental groups and covering spaces, the next cell in the Topology region.

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