Topology and Topological Space¶

Definition¶

A topology on a set $X$, often denoted by $\tau$, is a collection of subsets of $X$ that satisfies the following properties:

1. Both the empty set and the entire set $X$ belong to $\tau$.
2. Any union of elements in $\tau$ also belongs to $\tau$.
3. The intersection of any finite number of elements in $\tau$ remains in $\tau$.

The pair $(X,\tau)$ is called a topological space.

Interpretation¶

A topology on $X$ is a choice of which subsets count as open. It describes which points are close to which without using a distance: instead of balls of radius $\varepsilon$ one works with the open sets directly, and the three axioms keep the properties that open sets of a metric space always have.

$\tau=\{\varnothing,\{a\},\{a,b\},X\}$ is a topology on $X=\{a,b,c\}$$\{a,b,c\}$: it contains $\varnothing$ and $X$ and is closed under unions and finite intersections, e.g. $\{a\}\cup\{a,b\}=\{a,b\}$ and $\{a\}\cap\{a,b\}=\{a\}$$\{a\}$. The members of $\tau$ are the open sets. The only open set containing $c$ is $X$, so in this topology $c$ cannot be separated from $a$ or $b$ by an open set.

Arbitrary unions are allowed, but only finite intersections, because infinitely many open sets can shrink to a set that is not open: in $\mathbb{R}$ the intersection of the open intervals $(-\tfrac{1}{n},\tfrac{1}{n})$ is $\{0\}$, whose boundary point $0$ belongs to it. Formally, $\tau$ is a subset of the power set of $X$, not a subset of $X$.

Every metric space gives a topology (Key Results), but not every topology comes from a metric. In a metric space every one-point set is closed, but in the figure $\{a\}$ is not closed, since its complement $\{b,c\}$ is not in $\tau$.

This idea returns later: two metrics that give the same open sets are called topologically equivalent, and notions such as compactness are defined using open sets only.

Key Results¶

• Metric spaces are topological spaces. Every metric space induces a topology where open sets are unions of open metric balls.

Examples¶

• The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, i.e., $\tau=\{\varnothing, X\}$$\{\varnothing ,X\}$.

Prerequisites¶

• Subsets
• Empty Set
• Union of Sets
• Intersection of Sets
• Finite and Infinite Sets

Related Concepts¶

• Open and Closed Sets in a Metric Space

Sources¶

• Math III: Analysis · Lecture 5

Aliases: topological space Symbols: epsilon, intersection, real numbers, tau, union

Last updated: 20 October 2026 03:02