The Timpson Lyceum · General Topology (Point Set Topology)

Subspace & Product Topologies – Lecture 12 (Topology)

Overview

This lecture opens the chapter on constructing new topologies from old ones — rather than building a topology from scratch every time, we inherit one from a space we already understand. The first construction is the subspace topology, which lets any subset $Y \subseteq X$ borrow a topology from $X$ by intersecting $X$'s open sets with $Y$. This is what makes it easy to talk about topology on shapes like a circle, a torus, or any surface we can embed in $\mathbb{R}^2$ or $\mathbb{R}^3$. The second construction is the product topology, which builds a topology on a Cartesian product $X \times Y$ from the topologies on $X$ and $Y$ separately. Together these give us a vocabulary for spaces such as the unit square, the annulus, and the torus, built out of pieces we already know.

Key Concepts

Subspace topology. Let $(X, \tau)$ be a topological space and $Y \subseteq X$. The subspace topology on $Y$ is $\tau_Y = \{\, U \cap Y : U \in \tau \,\}.$ A set $V \subseteq Y$ is open in $Y$ if and only if $V \in \tau_Y$ — that is, $V = U \cap Y$ for some $U$ open in $X$. Each subset of $X$ inherits its own (possibly different) subspace topology.

• Open in $Y$ vs. open in $X$ are different. A set open in $Y$ need not be open in $X$ (intersecting with $Y$ can drag in boundary points of $Y$), and a set open in $X$ need not be open in $Y$ (it may contain points outside $Y$).
• $\tau_Y$ really is a topology. The three axioms are inherited from $\tau$:
  • $\varnothing = \varnothing \cap Y \in \tau_Y$ and $Y = X \cap Y \in \tau_Y$ (using $\varnothing, X \in \tau$).
  • Finite intersections: if $V_i = U_i \cap Y$, then $\bigcap_{i=1}^n V_i = \left(\bigcap_{i=1}^n U_i\right) \cap Y$, and a finite intersection of $X$-open sets is $X$-open.
  • Arbitrary unions: if $V_\alpha = U_\alpha \cap Y$, then $\bigcup_\alpha V_\alpha = \left(\bigcup_\alpha U_\alpha\right) \cap Y$, and an arbitrary union of $X$-open sets is $X$-open. (Both identities are just distributing $\cap Y$ in or out.)

Standard subspace topology. If $Y \subseteq \mathbb{R}^n$, the standard topology on $Y$ is the subspace topology inherited from the standard topology on $\mathbb{R}^n$ (whose open sets are unions of open balls / open intervals).

Closed sets in a subspace. A set $D \subseteq Y$ is closed in $Y$ iff its complement $Y \setminus D$ is open in $Y$. The defining-by-intersection property carries over to closed sets: $D \text{ is closed in } Y \iff D = C \cap Y \text{ for some } C \text{ closed in } X.$ As with open sets, "closed in $Y$" and "closed in $X$" are genuinely different — a set can be closed in $Y$ while being neither open nor closed in $X$.

Basis for a subspace. If $\mathcal{B}$ is a basis for $\tau$, then $\mathcal{B}_Y = \{\, B \cap Y : B \in \mathcal{B} \,\}$ is a basis for $\tau_Y$. (Recall a basis must cover every point, and for any point in the overlap of two basis elements there must be a third basis element inside the overlap containing that point.) So we never need the whole topology of $X$ — just its basis, intersected with $Y$.

Cartesian product of sets. $A \times B$ is the set of all ordered pairs $(a,b)$ with $a \in A$, $b \in B$. Crossing raises dimension: e.g. $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$. Pictorially, $A \times B$ is the "rectangle" swept out by pairing every point of $A$ with all of $B$.

Product topology. Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. The product topology on $X \times Y$ is generated by the basis $\mathcal{B} = \{\, U \times V : U \in \tau_X,\ V \in \tau_Y \,\}.$

• This is only a basis, not the full topology. Not every open set in $X \times Y$ is a product of two open sets — but every open set is a union of such products. (An L-shaped region in $\mathbb{R}^2$ is open yet is not a single $U \times V$.)
• Products distribute over intersections but not unions: $(A_1 \cap A_2) \times (B_1 \cap B_2) = (A_1 \times B_1) \cap (A_2 \times B_2)$, whereas $(A_1 \times B_1) \cup (A_2 \times B_2) \neq (A_1 \cup A_2) \times (B_1 \cup B_2)$ in general (the right side picks up extra "corner" pieces).
• Basis from bases. It suffices to use basis elements: if $\mathcal{B}_X$, $\mathcal{B}_Y$ are bases for $X$, $Y$, then $\{\, B_1 \times B_2 : B_1 \in \mathcal{B}_X,\ B_2 \in \mathcal{B}_Y \,\}$ is also a basis for the product topology.

Standard example spaces. With $I = [0,1]$, $S^1$ the circle, $D$ the disc:

• $I \times I$ = the unit square.
• $S^1 \times I$ = the (closed) annulus (a "washer"); $S^1 \times (0,1)$ = the open annulus.
• $S^1 \times S^1$ = the torus (surface of a donut).
• $S^1 \times D$ = the solid torus.

Worked Example

Goal: Determine the subspace topology that the integers $\mathbb{Z}$ inherit from the standard topology on $\mathbb{R}$.

The recipe is $\tau_{\mathbb{Z}} = \{\, U \cap \mathbb{Z} : U \text{ open in } \mathbb{R} \,\}$. Test it on individual open intervals, then see what subsets of $\mathbb{Z}$ are reachable:

Around any single integer $n$ we can place a small interval $\left(n - \tfrac12,\, n + \tfrac12\right)$ whose intersection with $\mathbb{Z}$ is exactly $\{n\}$. So every singleton $\{n\}$ is open in $\mathbb{Z}$. Taking unions of these intervals, any subset of $\mathbb{Z}$ can be realized as $U \cap \mathbb{Z}$. Therefore $\tau_{\mathbb{Z}} = \mathcal{P}(\mathbb{Z}) \quad\text{— the discrete topology.}$ The integers, sitting inside $\mathbb{R}$, inherit the discrete topology: every subset is open.

Summary

• The subspace topology on $Y \subseteq X$ is $\tau_Y = \{U \cap Y : U \in \tau\}$; this lets any subset borrow a topology, and it satisfies all three topology axioms by inheritance.
• "Open in $Y$" and "open in $X$" (and likewise "closed") are distinct notions — intersecting with $Y$ can add or remove boundary points.
• A set is closed in $Y$ iff it is $C \cap Y$ for some $C$ closed in $X$; and a basis for $X$ intersected with $Y$ gives a basis for $Y$.
• The product topology on $X \times Y$ is generated by the basis of all products $U \times V$ of open sets; not every open set is such a product, but every open set is a union of them — and using just basis elements $B_1 \times B_2$ suffices.
• Products distribute over intersections but not over unions, so be careful when combining product sets.
• These constructions yield the standard model spaces: the unit square $I \times I$, the annulus $S^1 \times I$, and the torus $S^1 \times S^1$.