The Timpson Lyceum · General Topology (Point Set Topology)

Open Sets – Lecture 6 (Topology)

Open Sets – Lecture 6

Overview

This lecture cements the definition of a topology and the language of open sets, then puts that definition to work. We review the three axioms a collection of subsets must satisfy to be a topology, revisit the trivial and discrete topologies, and build the central running example of the course: the finite complement topology on $\mathbb{R}$ . We then prove that this construction really is a topology, introduce the notions of finer and coarser topologies, define a topological space and a neighborhood, and prove an "open iff locally open" characterization. Two homework results close the lecture: the discrete and finite complement topologies coincide on a finite set but differ on an infinite one, and a quick characterization of the discrete topology in terms of singletons.

Key Concepts

Topology. Let $X$ be a set and let $\tau$ be a collection of subsets of $X$ . Then $\tau$ is a topology on $X$ if it satisfies three properties:

$\varnothing \in \tau$ and $X \in \tau$ .
Finite intersections: if $U_1, U_2, \dots, U_n \in \tau$ , then $\bigcap_{i=1}^{n} U_i \in \tau$ .
Arbitrary unions: for any (possibly infinite) collection $\{U_\alpha\} \subseteq \tau$ , the union $\bigcup_\alpha U_\alpha \in \tau$ .

Open set. A set is open exactly when it is a member of the topology $\tau$ . "Open" is just shorthand for "in $\tau$ ," used so we don't keep saying "in the topology."

Power set. $\mathcal{P}(X)$ is the set of all subsets of $X$ . For example, $\mathcal{P}(\{a,b\}) = \{\,\varnothing,\ \{a\},\ \{b\},\ \{a,b\}\,\}.$ Every topology on $X$ is a subset of $\mathcal{P}(X)$ , since a topology is built from subsets of $X$ .

Trivial topology. $\tau = \{\varnothing, X\}$ . The smallest possible topology on any set.

Discrete topology. $\tau = \mathcal{P}(X)$ , every subset is open. The largest possible topology. It satisfies all three axioms because any intersection or union of subsets of $X$ is again a subset of $X$ .

Finite complement topology $\mathbb{R}_{FC}$ (the lecture's central example). On $\mathbb{R}$ , declare a set open when its complement is finite, or when the set is empty: $\tau = \{\, U \subseteq \mathbb{R} : \mathbb{R} \setminus U \text{ is finite} \,\} \cup \{\varnothing\}.$

The complement of a set is everything not in it. It is always taken relative to the ambient set.
Example: $U = \mathbb{R} \setminus \{0, \tfrac{\sqrt{2}}{2}, \pi\}$ is open — the real line with three points punched out — because its complement $\{0, \tfrac{\sqrt{2}}{2}, \pi\}$ is finite.
A ray like $[0, \infty)$ is not open here: its complement $(-\infty, 0)$ is infinite.
$\varnothing$ must be included by fiat, since its complement is all of $\mathbb{R}$ , which is infinite.
Intuition for the proof: as you union sets together they grow, so the missing points (the complement) can only shrink; as you intersect sets they shrink, so the complement grows. Unioning the sets corresponds to intersecting their complements, and intersecting the sets to unioning their complements.

Topological space. The ordered pair $(X, \tau)$ — a set together with a topology on it. (Compare a sample space in probability: a set bundled with structure.)

Finer and coarser. Given two topologies $\tau_1$ and $\tau_2$ on the same set $X$ :

If $\tau_1 \subseteq \tau_2$ , then $\tau_2$ is finer than $\tau_1$ and $\tau_1$ is coarser than $\tau_2$ . Finer = more open sets = broken into smaller pieces.
If $\tau_1 \subsetneq \tau_2$ (strict subset), then $\tau_2$ is strictly finer and $\tau_1$ is strictly coarser.

Neighborhood. Given $(X, \tau)$ and a point $x \in X$ , any open set $U_x \in \tau$ with $x \in U_x$ is a neighborhood of $x$ . The notation $U_x$ always means an open set containing the point $x$ .

"Open iff locally open" theorem. Let $(X, \tau)$ be a topological space and $A \subseteq X$ . Then $A$ is open if and only if for every $x \in A$ there exists an open set $U_x \in \tau$ with $x \in U_x \subseteq A$ . (Forward direction is immediate: take $U_x = A$ . Reverse direction unions all the $U_x$ to recover $A$ , via the union lemma.)

Singleton characterization of the discrete topology. $\tau = \mathcal{P}(X)$ if and only if $\{x\} \in \tau$ for every $x \in X$ . Once every singleton is open, any subset is a union of singletons and therefore open.

Discrete vs. finite complement. On a finite set, $\mathcal{P}(X) = \mathbb{R}_{FC}$ -style finite complement topology — they are the same. On an infinite set they differ: a singleton $\{a\}$ is open in the discrete topology but not in the finite complement topology, since its complement $X \setminus \{a\}$ is infinite.

Worked Example

Claim: The finite complement topology $\mathbb{R}_{FC}$ is a topology — verify the finite-intersection axiom for nonempty open sets.

Let $U_1, \dots, U_n \in \mathbb{R}_{FC}$ with no $U_i = \varnothing$ . Each is open, so each has a finite complement: there is a finite set $F_i$ with $U_i = \mathbb{R} \setminus F_i$ . The key set identity (a generalized De Morgan law, proved by writing membership as a logical conjunction and factoring out the negation) is:

\begin{aligned} \bigcap_{i=1}^{n} U_i &= \bigcap_{i=1}^{n} \left( \mathbb{R} \setminus F_i \right) \\ &= \mathbb{R} \setminus \bigcup_{i=1}^{n} F_i. \end{aligned}

A finite union of finite sets $\bigcup_{i=1}^{n} F_i$ is finite, so $\mathbb{R} \setminus \bigcup F_i$ has a finite complement and is therefore open — i.e. $\bigcap U_i \in \mathbb{R}_{FC}$ . (If some $U_i = \varnothing$ , the whole intersection is $\varnothing$ , which is in $\tau$ by definition. The $\varnothing, X$ and arbitrary-union axioms are checked similarly, the union argument relying on the fact that unioning sets only shrinks the complement.) $\blacksquare$

Summary

A topology $\tau$ on $X$ is a collection of subsets containing $\varnothing$ and $X$ , closed under finite intersections and arbitrary unions; its members are the open sets.
The trivial topology $\{\varnothing, X\}$ and the discrete topology $\mathcal{P}(X)$ are the smallest and largest topologies on any set.
The finite complement topology declares a set open iff its complement is finite (or it is empty); it is the recurring example of the course, proven a topology via the identity $\bigcap(\mathbb{R} \setminus F_i) = \mathbb{R} \setminus \bigcup F_i$ .
A topological space is the pair $(X, \tau)$ ; $\tau_1 \subseteq \tau_2$ makes $\tau_2$ finer and $\tau_1$ coarser, with strict inclusion giving strictly finer/coarser.
A set $A$ is open iff every point of $A$ has a neighborhood contained in $A$ — open-ness is a local, point-by-point property.
The discrete and finite complement topologies agree on finite sets but differ on infinite sets, where singletons have infinite complements and so fail to be finite-complement-open.