Subspace, Product & Quotient Topologies

Three canonical ways to build new spaces from old ones.

The idea

The subspace, product, and quotient topologies equip the three standard constructions of new spaces from old — cutting out a subset, forming a Cartesian product, gluing points together — each with a topology. In each case a natural map is in play, and the topology is chosen so that this map is continuous.

For a subset $Y \subseteq X$ the map is the inclusion of $Y$ into $X$, and the subspace topology takes as the open sets of $Y$ the intersections $U \cap Y$ with $U$ open in $X$. A set can be open in $Y$ without being open in $X$: openness is relative to the ambient space.

For a product $X \times Z$ the maps are the two projections, and the product topology takes as a basis the rectangles $U \times V$ with $U$ open in $X$ and $V$ open in $Z$. On $\mathbb{R}^{2}$ this gives exactly the Euclidean topology.

For gluing, a surjection $q \colon X \to Q$ identifies points of $X$, and the quotient topology declares $W \subseteq Q$ open exactly when $q^{-1}(W)$ is open in $X$. Gluing the endpoints of $[0, 1]$ into a circle shows the effect: a small arc around the glued point is open, because it pulls back to the two half-neighborhoods $[0, \varepsilon)$ and $(1 - \varepsilon, 1]$, which are open in $[0, 1]$. In all three cases the topology is the extreme one that makes the map continuous: the fewest open sets for the subspace and the product, the most for the quotient.

Ways to work on it

• Walkthrough. The defining rule for each of the three constructions.
• Practice. Decide whether a set is open in a subspace, product, or quotient.
• Hardest. Count the open sets of a finite product and check that projections are continuous.

Not sure where to start? Take the ten-question placement test.