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MathidiaMathematics Academy
Topology
Lesson 15 of 75
Review

Product and quotient spaces

Objective. Construct spaces from identifications.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The product topology on X×Y is generated by the basis of sets U×V with U open in X and V open in Y, making projections continuous and giving the coarsest topology doing so. A quotient space X/~ is formed by an equivalence relation ~, with the quotient topology declaring U⊆X/~ open exactly when its preimage under the quotient map q:X→X/~ is open in X, which is the finest topology making q continuous. For example, identifying the endpoints of [0,1] (0~1) produces a quotient space homeomorphic to the circle S¹, since gluing the two ends of an interval creates a loop.

Worked example

Identify the quotient space obtained from the square [0,1]×[0,1] by gluing (0,y)~(1,y) for all y (identify left and right edges only).

Diagram of a square
  1. 1Gluing only the left and right edges of the square with matching y-coordinates
  2. 2This rolls the square into a tube shape
  3. 3No other identifications are made, so the top and bottom remain open edges
  4. 4The resulting quotient space is a cylinder

Answer: A cylinder

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Common mistake. Students think the quotient topology is just the subspace topology inherited from X, but quotient topologies are defined via preimages under the quotient map and can differ substantially; for instance collapsing [0,1] to a point by x~y for all x,y gives the trivial one-point space, not a subspace of [0,1].

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