The concept of an open set is the foundation on which all of topology rests. In a [topological space](/page/Topology), the open sets are the primitive data — the collection of subsets designated as "open" determines convergence, [continuity](/page/Continuity), compactness, connectedness, and every other topological property. In a metric space, the open sets are those that contain a ball around each of their points; this concrete picture provides the intuition, while the axiomatic formulation provides the generality.

This page develops the concept of an open set in depth: what open sets look like in concrete spaces, how the basic operations on open sets behave (and why they behave asymmetrically), the local structure that open sets provide through neighbourhoods, and the role of open sets as the building blocks for maps and covers. The abstract framework — topological spaces, continuity, compactness — is developed on the [Topology](/page/Topology) page; here we focus on the open sets themselves.

[motivation] ## Motivation ### The Metric Origin In a metric space $(X, d)$, the notion of "closeness" is quantified by the distance [function](/page/Function). A set $U \subseteq X$ is declared **open** if every point of $U$ has some room around it: for each $x \in U$, there exists $\varepsilon > 0$ such that the open ball $B(x, \varepsilon) = \{y \in X : d(x, y) < \varepsilon\}$ is entirely contained in $U$. This captures the intuition that an open set has no "edge points" — from any point in $U$, one can move a small distance in any direction and remain inside $U$. For instance, the open interval $(0, 1) \subset \mathbb{R}$ is open: any point $x \in (0,1)$ has $\varepsilon = \min(x, 1-x) > 0$ with $(x - \varepsilon, x + \varepsilon) \subseteq (0,1)$. The closed interval $[0, 1]$ is not open: the endpoints $0$ and $1$ have no $\varepsilon$-ball contained in $[0,1]$, since every ball around $0$ contains negative numbers. ### Why Axiomatise? Two observations motivate the passage from metric openness to the axiomatic definition. First, many important topologies — pointwise convergence on function spaces, the weak and weak* topologies in functional analysis — are not metrizable, yet the notion of "open set" still governs convergence and continuity. Second, even in metric spaces, the open sets satisfy three closure properties (empty set and whole space are open, arbitrary unions are open, finite intersections are open) that depend only on the *collection* of open sets, not on the specific values of the distance function. These three properties are the axioms of a [topology](/page/Topology), and they are sufficient to rebuild all of point-set topology without reference to any distance. ### The Asymmetry The axioms allow arbitrary unions of open sets to be open, but only *finite* intersections. This asymmetry is not a design choice — it reflects what actually happens in [metric spaces](/page/Metric%20Space). The intersection $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$ is a single point, which is not open in $\mathbb{R}$: no ball around $0$ fits inside $\{0\}$. But any union of open balls is open, because a point in the union belongs to at least one ball and therefore has its own sub-ball. This asymmetry has deep consequences: the theory of [closed sets](/page/Closed%20Set) (complements of open sets) has the dual asymmetry — arbitrary intersections of closed sets are closed, but infinite unions need not be. [/motivation]

## Definition

In a topological space, open sets are primitive — they are the elements of the topology. In a metric space, they are derived from the distance function.

[definition:Open Set In A Topological Space] Let $(X, \tau)$ be a [topological space](/page/Topology). A subset $U \subseteq X$ is **open** if $U \in \tau$. [/definition]

The axioms of a topology guarantee that $\varnothing$ and $X$ are open, that arbitrary unions of open sets are open, and that finite intersections of open sets are open.

In a metric space, the definition of openness is concrete and can be checked pointwise:

[definition:Open Set In A Metric Space] Let $(X, d)$ be a metric space. A subset $U \subseteq X$ is **open** (with respect to the metric topology) if for every $x \in U$, there exists $\varepsilon > 0$ such that \begin{align*} B(x, \varepsilon) := \{y \in X : d(x, y) < \varepsilon\} \subseteq U. \end{align*} [/definition]

The metric definition says that $U$ is open if and only if every point of $U$ is an **interior point** — it has a ball-shaped neighbourhood entirely inside $U$. The collection of all open sets in a metric space satisfies the topology axioms, so every metric space is canonically a topological space. Different metrics on the same set can produce different topologies, but two metrics that produce the same open sets are called **topologically equivalent**.

## Open Sets in Metric Spaces

The abstract axioms guarantee that open sets are closed under unions and finite intersections, but they say nothing about *what open sets look like* in any particular space. In metric spaces — especially in $\mathbb{R}$ and $\mathbb{R}^n$ — the open sets have concrete geometric descriptions.

### Open Balls

The most basic open set in a metric space is the open ball. Every open set is a union of open balls (since the open balls form a basis for the metric topology), so understanding open balls is the key to understanding all open sets.

[definition:Open Ball] Let $(X, d)$ be a metric space, $x \in X$, and $r > 0$. The **open ball** of radius $r$ centred at $x$ is \begin{align*} B(x, r) := \{y \in X : d(x, y) < r\}. \end{align*} [/definition]

Open balls are open sets: if $y \in B(x, r)$, then $\delta := r - d(x, y) > 0$, and the triangle inequality gives $B(y, \delta) \subseteq B(x, r)$ (for any $z$ with $d(y, z) < \delta$, we have $d(x, z) \le d(x, y) + d(y, z) < d(x, y) + \delta = r$). This argument — which uses only the triangle inequality — is the prototype for all "a set is open because each point has a ball inside it" arguments.

The geometry of open balls depends on the metric. In $\mathbb{R}^2$ with the Euclidean metric $d_2$, open balls are open discs. With the $\ell^1$ metric $d_1(x, y) = |x_1 - y_1| + |x_2 - y_2|$, open balls are open squares rotated $45°$. With the $\ell^\infty$ metric $d_\infty(x, y) = \max(|x_1 - y_1|, |x_2 - y_2|)$, open balls are open squares aligned with the axes. All three metrics produce the same open sets — the same topology — despite having different ball geometries.

### The Structure of Open Sets in $\mathbb{R}$

In the real line, the order structure provides a complete description of what open sets look like. The fundamental result is that every open set decomposes uniquely into a countable collection of disjoint open intervals:

[quotetheorem:623]