The Timpson Lyceum · General Topology (Point Set Topology)

Product and Quotient Space Topologies – Lecture 13 (Topology)

Product and Quotient Space Topologies – Lecture 13

Overview

This lecture continues the program of building new topologies out of old ones. It first finishes the product topology — reviewing its basis, warning about a common cross-product trap, and proving three theorems about how interiors, subspaces, and closed sets behave under products. It then introduces the quotient topology, the construction that lets us "glue" a space to itself to form new objects. We prove that a quotient map really does induce a topology, and we develop intuition through gluing examples that produce the circle, annulus, torus, Möbius band, Klein bottle, projective plane, and connected sums.

Key Concepts

Product topology. Given topological spaces $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$, the product topology on $X \times Y$ is generated by the basis $\mathcal{B} = \{\, U \times V : U \in \mathcal{T}_X,\; V \in \mathcal{T}_Y \,\}.$ A basis element is an open set of $X$ crossed with an open set of $Y$; every open set of $X \times Y$ is a union of such boxes.

Basis, not all open sets. Crossing two open sets always yields an open set, but the converse fails: a general open set of $X \times Y$ need not be expressible as one open box $U \times V$. For example $(A_1 \times B_1) \cup (A_2 \times B_2)$ is open yet cannot be written as a single product. (One may also restrict the generators to products of basis elements of $X$ and $Y$ — these still generate the same product topology.)

Cross-product algebra (the trap). Cross products distribute over intersection but not over union: $(A_1 \cap A_2)\times(B_1\cap B_2) = (A_1\times B_1)\cap(A_2\times B_2),$ whereas $(A_1\cup A_2)\times(B_1\cup B_2)$ is strictly larger than $(A_1\times B_1)\cup(A_2\times B_2)$ — the union of products misses the "off-diagonal" corner pieces.

Interior of a product. For $A\subseteq X$, $B\subseteq Y$: $\operatorname{Int}(A)\times\operatorname{Int}(B)=\operatorname{Int}(A\times B).$ The interior of a set is the union of all open subsets it contains (its largest open subset).

Subspace vs. product agree. Topologizing $A\subseteq X$ and $B\subseteq Y$ two ways gives the same topology on $A\times B$: (1) take the subspace topology induced from the product $X\times Y$, or (2) take the product of the subspace topologies on $A$ and $B$. These coincide because $(U\times V)\cap(A\times B)=(U\cap A)\times(V\cap B)$.

Product of closed sets is closed. If $C$ is closed in $X$ and $D$ is closed in $Y$, then $C\times D$ is closed in $X\times Y$. The proof shows the complement is open by rewriting it as $\big((X\setminus C)\times Y\big)\cup\big(X\times(Y\setminus D)\big)$, a union of open boxes.

Quotient topology. Let $(X,\mathcal{T})$ be a space, $A$ any set, and $p:X\to A$ a surjective map (the quotient map). Define $V\subseteq A$ to be open exactly when its preimage is open in $X$: $V \text{ open in } A \iff p^{-1}(V)\in\mathcal{T}.$ The resulting collection $\mathcal{T}_A$ is the quotient topology induced by $p$. Different quotient maps on the same $X,A$ can induce different topologies. (Recall: $p(x)=y$ makes $y$ the image of $x$; $p^{-1}(y)$ is the preimage, the set of all points mapping to $y$ — possibly several.)

Quotient by a partition / gluing. The maps of interest send $X$ onto a partition $X^*$ of $X$ (a family of disjoint subsets whose union is $X$). Each point lands in the one block containing it. Identifying points this way is "gluing." Two spaces are homeomorphic (the same, topologically) when there is a continuous bijection both ways — rubber-sheet deformations (stretching, never tearing or new gluing) preserve the type. Standard gluings of the filled square $I\times I$:

• Identify one pair of opposite edges (same orientation): annulus.
• Identify both pairs (same orientation): torus $T$.
• Identify one pair with a flip (opposite orientation): Möbius band — one side, one edge; needs 3 dimensions to embed.
• Identify both pairs, one with a flip: Klein bottle — one-sided, no inside/outside; needs 4 dimensions to embed without self-intersection.
• Disk with antipodal boundary points identified: projective plane.
• Disk with the whole boundary collapsed (taco fold, edge-only): sphere.

Connected sum. To "add" two surfaces $S_1 \mathbin{\#} S_2$: cut an open disk (a hole) out of each, then glue the two boundary circles together. Torus $\#$ torus yields the genus-2 (two-holed) surface. Cutting out an open set adds a new boundary edge; we always remove open sets so the boundary points survive to be glued.

Worked Example

Theorem. In the product topology, $\operatorname{Int}(A)\times\operatorname{Int}(B)=\operatorname{Int}(A\times B)$.

Proof. Show each side contains the other.

($\subseteq$) Since $\operatorname{Int}(A)$ is open in $X$ and $\operatorname{Int}(B)$ is open in $Y$, their product is a basic open set of $X\times Y$, and it sits inside $A\times B$. As $\operatorname{Int}(A\times B)$ is the union of all open subsets of $A\times B$:

($\supseteq$) Take any point $(x,y)\in\operatorname{Int}(A\times B)$. This interior is open, so some basic box contains the point inside $A\times B$:

Every point of $\operatorname{Int}(A\times B)$ lies in $\operatorname{Int}(A)\times\operatorname{Int}(B)$, giving the reverse inclusion. Both inclusions hold, so the sets are equal. $\blacksquare$

Summary

• The product topology on $X\times Y$ is generated by the boxes $U\times V$ ($U,V$ open); these are a basis, not all open sets — a general open set is only a union of boxes.
• Cross products distribute over intersection but not union; remember $(A_1\times B_1)\cup(A_2\times B_2)\subsetneq (A_1\cup A_2)\times(B_1\cup B_2)$.
• Three product theorems: interiors multiply ($\operatorname{Int}(A)\times\operatorname{Int}(B)=\operatorname{Int}(A\times B)$); the subspace-of-product and product-of-subspaces topologies agree; and a product of closed sets is closed.
• A quotient map $p:X\to A$ is a surjection, and $V$ is open in $A$ iff $p^{-1}(V)$ is open in $X$; this always defines a genuine topology (closed under arbitrary unions and finite intersections, containing $\varnothing$ and $A$).
• Quotients realize gluing: identifying edges of $I\times I$ builds the annulus, torus, Möbius band, and Klein bottle; collapsing a disk's boundary builds the sphere; antipodal identification builds the projective plane.
• The connected sum cuts a disk from each of two surfaces and glues the boundary circles, e.g. torus $\#$ torus $=$ the two-holed surface.