Subspace and Quotient Topology: Definitions with Examples

The subspace topology lets a subset inherit topology from its parent: a set is open in A exactly when it equals A intersected with an open set of X. The quotient topology goes the other way: after identifying points of X, a set in the quotient is open exactly when its preimage up in X is open. One restricts, the other glues.

These build nearly every interesting space: subspaces give circles in planes and intervals in lines; quotients make the circle from an interval by gluing endpoints, the torus from a square by gluing opposite edges, the Mobius strip by gluing with a twist.

Key Concepts

• Subspace topology: opens of A are {U ∩ A : U open in X}; openness in A is relative.
• Quotient map: surjection pi: X → Y with U open in Y iff pi^{-1}(U) open in X.
• Identification space: X modulo an equivalence relation, fusing equivalent points.
• Collapsing: identifying a whole subset to one point, e.g. a disc's boundary collapsed gives a sphere.
• Universal property: a map out of a quotient is continuous iff its composite with pi is.

Formulas

• Subspace opens: tau_A = {U ∩ A : U ∈ tau_X}.
• Quotient opens: U ⊆ Y open iff pi^{-1}(U) open in X.
• Circle as quotient: [0,1] with 0 ~ 1 ≅ S1 via t ↦ (cos 2 pi t, sin 2 pi t).

Worked Example

Subspace topology on A = [0,1] ⊂ R; gluing [0,1]'s endpoints gives the circle.

1. (1/2, 3/2) ∩ [0,1] = (1/2, 1] is open in A, though not open in R.
2. [0, 1/2) = (-1/2, 1/2) ∩ [0,1] is open in A; endpoints become interior from A's viewpoint.
3. pi(t) = (cos 2 pi t, sin 2 pi t) identifies 0 and 1, since pi(0) = pi(1).
4. A neighbourhood of (1,0) in S1 pulls back to [0,a) ∪ (b,1], open in the subspace [0,1].
5. pi is a continuous bijection from compact [0,1]/~ to Hausdorff S1, hence a homeomorphism.

Common Misconceptions

• Quotients can be violent: gluing can collapse regions to points or yield non-Hausdorff spaces; nice in, nasty out is possible.
• "Open" is meaningless without context: (1/2,1] is open in [0,1] but not in R.
• The equivalence relation is the whole data: gluing [0,1]'s endpoints gives a circle, identifying all points gives a single point.

Real-World Applications

• Geometry: torus, Klein bottle, Mobius strip, projective plane are quotients of squares/discs.
• Graphics: texture wrapping uses quotient identifications to map rectangles onto cylinders.
• Physics: string theory compactifies dimensions on torus quotients.

Subspaces inherit openness by intersection, so (1/2,1] is open in [0,1] though not in R; quotients declare opens by preimages, turning [0,1] with glued endpoints into the circle. Restricting and gluing are topology's two great construction tools.