Manifolds 4 | Quotient Spaces [dark version]

Quotient topology turns “collapsing” or “gluing” rules into a rigorous way to build new topological spaces. The core idea is simple: start with a topological space (X, T) and an equivalence relation ~ on X, then form the set of equivalence classes X/~, and define a topology on that quotient set so that the natural projection map q: X → X/~ is continuous. This matters because many important spaces in topology—like Möbius strips and projective spaces—are created by identifying points according to a rule, and quotient topology is the standard language for making those identifications mathematically precise.

The construction begins with equivalence classes: each point x in X belongs to a “box” [x] containing all points y that are equivalent to x. These boxes partition X, but the partition alone doesn’t yet define a topology on the collection of boxes. To do that, one uses the canonical projection q, which sends each point x to its equivalence class [x]. Open sets in the quotient space are then defined by compatibility with openness in the original space: a subset U of X/~ is declared open exactly when its preimage q⁻¹(U) is open in X. Because preimages under q interact cleanly with unions and intersections, the quotient topology satisfies the axioms of a topology, yielding a well-defined topology on X/~.

A concrete example is the Möbius strip built from a rectangle by gluing with a twist. The setup treats the rectangle as a product of a closed interval and an open interval: X = [0,1] × (−1,1). The gluing identifies points on the left boundary with points on the right boundary, but with a flip: (0, s) is identified with (1, −s). Under this equivalence relation, the quotient space becomes the Möbius strip. Determining whether a particular region of the resulting strip is open again uses the preimage test: a set is open in the Möbius strip precisely when its preimage in the original rectangle splits into open pieces that were open before the identification.

The same quotient-topology mechanism is then pointed toward projective space. Projective space P^n(R) is described as the set of one-dimensional subspaces of R^{n+1}, which can be visualized (for n = 1) as all lines through the origin in R^2. While the set of lines is straightforward to define, the key remaining step is choosing the topology induced by the original space—exactly the role quotient topology plays. The discussion sets up that projective space will be constructed using quotient topology in a subsequent segment, tying the abstract quotient method directly to a major class of geometric spaces.