Topological Spaces and Open Sets

A topological space is the foundational object of study in point-set topology. It abstracts the notion of "nearness" and "continuity" from metric spaces to a far more general setting, by axiomatizing the concept of open sets.

The Definition of a Topology

Definition1.1Topology on a Set

Let $X$ be a set. A topology on $X$ is a collection $\tau \subseteq \mathcal{P}(X)$ of subsets of $X$ satisfying the following axioms:

Empty set and whole space: $\emptyset \in \tau$ and $X \in \tau$ .
Arbitrary unions: If $\{U_\alpha\}_{\alpha \in A}$ is any family of sets in $\tau$ , then $\bigcup_{\alpha \in A} U_\alpha \in \tau$ .
Finite intersections: If $U_1, U_2, \ldots, U_n \in \tau$ , then $\bigcap_{i=1}^n U_i \in \tau$ .

The pair $(X, \tau)$ is called a topological space. The elements of $\tau$ are called open sets.

The asymmetry between axioms (2) and (3) is essential: arbitrary intersections of open sets need not be open. For instance, in $\mathbb{R}$ with the standard topology, $\bigcap_{n=1}^{\infty} (-1/n, 1/n) = \{0\}$ , which is not open.

Fundamental Examples

ExampleDiscrete and Indiscrete Topologies

Let $X$ be any set.

The discrete topology on $X$ is $\tau_{\text{disc}} = \mathcal{P}(X)$ , the collection of all subsets of $X$ . Every subset is open.
The indiscrete (trivial) topology on $X$ is $\tau_{\text{ind}} = \{\emptyset, X\}$ . Only the empty set and the whole space are open.

These represent the two extreme topologies on $X$ . For any topology $\tau$ on $X$ , we have $\tau_{\text{ind}} \subseteq \tau \subseteq \tau_{\text{disc}}$ .

ExampleStandard Topology on the Real Line

The standard (Euclidean) topology on $\mathbb{R}$ is defined by declaring a set $U \subseteq \mathbb{R}$ to be open if for every $x \in U$ , there exists $\epsilon > 0$ such that $(x - \epsilon, x + \epsilon) \subseteq U$ .

Equivalently, $U$ is open if and only if $U$ is a union of open intervals $(a, b)$ . The open intervals form a basis for this topology.

ExampleCofinite Topology

Let $X$ be any set. The cofinite topology (or finite complement topology) on $X$ is: $\tau_{\text{cof}} = \{U \subseteq X : X \setminus U \text{ is finite}\} \cup \{\emptyset\}.$

To verify axiom (2): if $\{U_\alpha\}$ are cofinite open sets, then $X \setminus \bigcup U_\alpha = \bigcap (X \setminus U_\alpha)$ , which is an intersection of finite sets, hence finite. For axiom (3): $X \setminus \bigcap_{i=1}^n U_i = \bigcup_{i=1}^n (X \setminus U_i)$ , which is a finite union of finite sets, hence finite.

Comparing Topologies

Definition1.2Coarser and Finer Topologies

Let $\tau_1$ and $\tau_2$ be two topologies on the same set $X$ .

We say $\tau_1$ is coarser (or weaker) than $\tau_2$ if $\tau_1 \subseteq \tau_2$ .
We say $\tau_1$ is finer (or stronger) than $\tau_2$ if $\tau_1 \supseteq \tau_2$ .

The collection of all topologies on $X$ forms a partially ordered set under inclusion.

RemarkLattice of Topologies

The collection of all topologies on $X$ actually forms a complete lattice. Given any family $\{\tau_\alpha\}$ of topologies on $X$ , the intersection $\bigcap_\alpha \tau_\alpha$ is again a topology on $X$ (the meet). The join is the coarsest topology containing all $\tau_\alpha$ , which is the intersection of all topologies containing $\bigcup_\alpha \tau_\alpha$ .

Open Sets and Neighborhoods

Definition1.3Neighborhood

Let $(X, \tau)$ be a topological space and $x \in X$ . A neighborhood of $x$ is a set $N \subseteq X$ such that there exists an open set $U \in \tau$ with $x \in U \subseteq N$ .

The collection $\mathcal{N}(x)$ of all neighborhoods of $x$ is called the neighborhood system (or neighborhood filter) of $x$ .

The neighborhood system satisfies the following properties, which in fact characterize topologies:

$X \in \mathcal{N}(x)$ and $x \in N$ for every $N \in \mathcal{N}(x)$ .
If $N_1, N_2 \in \mathcal{N}(x)$ , then $N_1 \cap N_2 \in \mathcal{N}(x)$ .
If $N \in \mathcal{N}(x)$ and $N \subseteq M$ , then $M \in \mathcal{N}(x)$ .
If $N \in \mathcal{N}(x)$ , there exists $M \in \mathcal{N}(x)$ such that $N \in \mathcal{N}(y)$ for all $y \in M$ .

Definition1.4Interior

Let $(X, \tau)$ be a topological space and $A \subseteq X$ . The interior of $A$ , denoted $\operatorname{int}(A)$ or $A^\circ$ , is the largest open set contained in $A$ : $\operatorname{int}(A) = \bigcup \{U \in \tau : U \subseteq A\}.$

A point $x$ is an interior point of $A$ if and only if $A$ is a neighborhood of $x$ .

RemarkInterior as Idempotent Operator

The interior operator satisfies the Kuratowski interior axioms:

$\operatorname{int}(X) = X$ .
$\operatorname{int}(A) \subseteq A$ .
$\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)$ (idempotence).
$\operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B)$ .

A topology on $X$ can equivalently be defined by specifying an interior operator satisfying these axioms.