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] The proof constructs, for each point $x \in U$, the maximal open interval containing $x$ and contained in $U$, using the completeness of $\mathbb{R}$ (the [supremum and infimum](/page/Supremum%20and%20Infimum) exist). Maximality forces distinct intervals to be disjoint, and countability follows from the density of $\mathbb{Q}$: each interval contains a distinct rational, and $\mathbb{Q}$ is countable. This result is specific to $\mathbb{R}$ and relies on two features of the real line: the order structure (which makes intervals the natural building blocks) and the separability (which forces countability). In $\mathbb{R}^2$, open sets have no analogous decomposition into disjoint "simple" pieces — an open connected subset of $\mathbb{R}^2$ can have arbitrarily complicated topology (it can be simply connected, have finitely many holes, or even be a fractal-like region). [example:Decomposition Of A Union Of Intervals] The open set $U = (0, 1) \cup (1, 3) \cup (5, \infty) \subset \mathbb{R}$ is already written as a disjoint union of three open intervals. This is its maximal-interval decomposition: $(0,1)$, $(1,3)$, and $(5, \infty)$ are each maximal (they cannot be extended within $U$ because the points $1$, $3$, and $5$ are not in $U$, and there is nothing between $3$ and $5$ in $U$). A less obvious example: $U = \mathbb{R} \setminus \mathbb{Z} = \bigcup_{n \in \mathbb{Z}} (n, n+1)$. This is a countably infinite disjoint union of open intervals, each of length $1$. [/example] [example:Cantor Set Complement] The complement of the Cantor set $C \subset [0,1]$ in $[0,1]$ is an open set in the subspace topology. Its maximal-interval decomposition is the union of the "middle thirds" removed at each stage of the Cantor construction: $(1/3, 2/3)$, $(1/9, 2/9)$, $(7/9, 8/9)$, and so on — countably many disjoint open intervals whose total length sums to $1$ (since $\mathcal{L}^1(C) = 0$). This example shows that the intervals in the decomposition can be dense in $[0,1]$, with endpoints accumulating everywhere. [/example] ### Open Sets in $\mathbb{R}^n$ and General Metric Spaces In $\mathbb{R}^n$ with any of the standard metrics ($\ell^1$, $\ell^2$, $\ell^\infty$), an open set is a union of open balls. Unlike in $\mathbb{R}$, there is no canonical decomposition into disjoint "simple" pieces. However, every open set in $\mathbb{R}^n$ can be written as a *countable* union of open balls (with rational centres and rational radii), because $\mathbb{R}^n$ is second countable — the balls with rational centres and radii form a countable basis. More generally, in any [separable](/page/Separable) metric space, every open set is a countable union of open balls. This follows from the [equivalence of separability and second countability in metrizable spaces](/theorems/545), since a second-countable space has a countable basis and every open set is a union of basis elements. ## Operations on Open Sets The topology axioms — arbitrary unions, finite intersections, and the requirement that $\varnothing$ and $X$ be open — govern how open sets combine. Understanding why these axioms take the form they do, and what happens when the restrictions are violated, is essential for working with open sets. ### The Union-Intersection Asymmetry The axioms are asymmetric: arbitrary unions of open sets are open, but only *finite* intersections are guaranteed to be open. In a metric space, the reason is transparent. **Why arbitrary unions work.** If $\{U_i\}_{i \in I}$ are open and $x \in \bigcup_i U_i$, then $x \in U_j$ for some $j$. Since $U_j$ is open, there exists $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U_j \subseteq \bigcup_i U_i$. So $\bigcup_i U_i$ is open. **Why finite intersections work.** If $U_1, \ldots, U_n$ are open and $x \in U_1 \cap \cdots \cap U_n$, then for each $k$ there exists $\varepsilon_k > 0$ with $B(x, \varepsilon_k) \subseteq U_k$. Set $\varepsilon = \min(\varepsilon_1, \ldots, \varepsilon_n) > 0$ (the minimum of finitely many positive numbers is positive). Then $B(x, \varepsilon) \subseteq U_k$ for all $k$, so $B(x, \varepsilon) \subseteq \bigcap_k U_k$. The critical step in the intersection argument is that the minimum of *finitely many* positive numbers is positive. For infinitely many, this can fail — and it does. ### Why Infinite Intersections Fail The failure of infinite intersections to preserve openness is not an edge case; it is a common and important phenomenon. [example:Intersection Collapsing To A Point] In $\mathbb{R}$, set $U_n = (-1/n, 1/n)$ for $n \in \mathbb{N}$. Each $U_n$ is open. The intersection is \begin{align*} \bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}, \end{align*} which is not open: no ball $(-\varepsilon, \varepsilon)$ fits inside the singleton $\{0\}$ for $\varepsilon > 0$. The "radii" $1/n$ shrink to zero, so the minimum of infinitely many positive numbers is zero — the mechanism that preserved openness in the finite case collapses. [/example] [example:Intersection Producing A Closed Set] In $\mathbb{R}^2$, define $U_n = \{(x, y) : |y| < 1/n\}$ — the open horizontal strip of height $2/n$ centred on the $x$-axis. Each $U_n$ is open. The intersection $\bigcap_{n=1}^\infty U_n = \{(x, 0) : x \in \mathbb{R}\}$ is the $x$-axis, which is closed (not open) in $\mathbb{R}^2$. This example shows that a countable intersection of open sets can be a closed set of lower dimension. [/example] A countable intersection of open sets is called a $G_\delta$ **set** (from the German *Gebiet*, "region," and *Durchschnitt*, "intersection"). Not every $G_\delta$ set is open, and not every $G_\delta$ set is closed — the class of $G_\delta$ [sets](/page/Set) is strictly larger than both. For instance, the set of irrationals $\mathbb{R} \setminus \mathbb{Q}$ is a $G_\delta$ set (write $\mathbb{R} \setminus \mathbb{Q} = \bigcap_{q \in \mathbb{Q}} (\mathbb{R} \setminus \{q\})$, a countable intersection of open sets), but it is neither open nor closed. $G_\delta$ sets play an important role in descriptive set theory and in results like the [Baire category theorem](/theorems/630), where the intersection of countably many *dense* open sets is shown to be dense in complete metric spaces. ## Neighbourhoods and Local Structure The global axioms — arbitrary unions, finite intersections — describe the algebraic structure of the collection of open sets. But much of topology is *local*: convergence, continuity at a point, and the closure operator are all defined by what happens "near" a given point. The concept that mediates between the global (open sets) and the local (behaviour near a point) is the **neighbourhood**. ### Neighbourhood Systems A neighbourhood of a point is any set that contains an open set around the point. The collection of all neighbourhoods of $x$ — the neighbourhood system — encodes everything about the local topology at $x$. [definition:Neighbourhood] Let $(X, \tau)$ be a topological space and $x \in X$. A **neighbourhood** of $x$ is any set $N \subseteq X$ such that there exists an open set $U \in \tau$ with $x \in U \subseteq N$. The collection $\mathcal{N}(x) := \{N \subseteq X : \exists\, U \in \tau,\; x \in U \subseteq N\}$ is the **neighbourhood system** (or **neighbourhood filter**) of $x$. [/definition] A neighbourhood need not be open — a closed ball $\overline{B}(x, r) = \{y : d(x,y) \le r\}$ in a metric space is a neighbourhood of $x$ (it contains the open ball $B(x, r)$), but it is not open (in general). However, every neighbourhood *contains* an open neighbourhood, and a set is open if and only if it is a neighbourhood of every one of its points: $U \in \tau$ if and only if $U \in \mathcal{N}(x)$ for all $x \in U$. This is the link between the global and local viewpoints. The neighbourhood system $\mathcal{N}(x)$ is a **filter**: it is nonempty ($X \in \mathcal{N}(x)$), closed under finite intersections, closed upward (if $N \in \mathcal{N}(x)$ and $N \subseteq M$, then $M \in \mathcal{N}(x)$), and does not contain $\varnothing$. The topology on $X$ is completely determined by the neighbourhood systems: the axioms for neighbourhood systems (due to Hausdorff) provide an equivalent foundation for topology that takes "neighbourhood" rather than "open set" as primitive. ### Local Bases In practice, one rarely works with the entire neighbourhood system. Instead, one uses a smaller generating collection — a **local basis** — from which all neighbourhoods can be recovered. [definition:Local Basis] A **local basis** (or **neighbourhood basis**) at $x$ is a subcollection $\mathcal{B}(x) \subseteq \mathcal{N}(x)$ such that for every $N \in \mathcal{N}(x)$, there exists $B \in \mathcal{B}(x)$ with $B \subseteq N$. [/definition] In a metric space, the open balls $\{B(x, 1/n) : n \in \mathbb{N}\}$ form a countable local basis at every point $x$. A topological space in which every point has a countable local basis is called **first countable**; all metric spaces are first countable. First countability is the condition under which [sequences](/page/Sequence) suffice to detect all topological properties — in a first-countable space, a set is open if and only if no sequence outside it converges to a point inside it. ### Characterisation of Closure and Interior The closure and interior of a set can be characterised in terms of open sets and neighbourhoods. These characterisations are the working definitions in most proofs. A point $x$ belongs to the closure $\overline{A}$ if and only if every open set containing $x$ meets $A$ — equivalently, every neighbourhood of $x$ intersects $A$. A point $x$ belongs to the interior $A^\circ$ if and only if some open set containing $x$ is contained in $A$ — equivalently, $A$ is a neighbourhood of $x$. [example:Closure Via Open Sets] In $\mathbb{R}$, the closure of $\mathbb{Q}$ is $\overline{\mathbb{Q}} = \mathbb{R}$: for any $x \in \mathbb{R}$ and any open interval $(x - \varepsilon, x + \varepsilon)$ around $x$, the density of $\mathbb{Q}$ guarantees a rational in this interval. Conversely, the interior of $\mathbb{Q}$ is $\mathbb{Q}^\circ = \varnothing$: no open interval is contained in $\mathbb{Q}$, since every open interval contains irrationals. This shows that $\mathbb{Q}$ is "topologically small" (nowhere dense after taking the closure of the complement), despite being algebraically dense. [/example] ## Open Maps Continuity is defined by the condition that preimages of open sets are open. There is a complementary but fundamentally different condition on a function: that *images* of open sets are open. ### Definition and Comparison with Continuity A continuous function pulls open sets *backward* (preimages of open sets are open). An open map pushes open sets *forward* (images of open sets are open). These are independent conditions: a function can be one without being the other. [definition:Open Map] A function $f: X \to Y$ between topological spaces is an **open map** if for every open set $U \subseteq X$, the image $f(U)$ is open in $Y$. [/definition] [example:Projection As Open Map] The projection $\pi_1: \mathbb{R}^2 \to \mathbb{R}$ defined by $\pi_1(x, y) = x$ is both continuous and open. It is continuous because the preimage of an open interval $(a, b)$ is the open strip $(a, b) \times \mathbb{R}$, which is open in $\mathbb{R}^2$. It is open because the image of a basic open set $U \times V$ (with $U, V$ open in $\mathbb{R}$) is $U$ if $V \neq \varnothing$, which is open. However, $\pi_1$ is not a closed map: the image of the closed set $\{(x, y) : xy = 1\}$ (a hyperbola) under $\pi_1$ is $\mathbb{R} \setminus \{0\}$, which is not closed. [/example] [example:Continuous But Not Open] The inclusion $\iota: [0, 1] \hookrightarrow \mathbb{R}$ (with the subspace topology on $[0,1]$) is continuous but not open. The set $[0, 1/2)$ is open in the subspace $[0,1]$ (since $[0, 1/2) = (-1, 1/2) \cap [0,1]$), but its image in $\mathbb{R}$ is $[0, 1/2)$, which is not open in $\mathbb{R}$. [/example] ### When Continuous Bijections Are Open A bijection $f: X \to Y$ is a homeomorphism if and only if it is continuous and open (equivalently, continuous and closed). The [Topological Inverse Function Theorem](/theorems/318) provides a sufficient condition: a continuous bijection from a compact space to a Hausdorff space is automatically open. Outside this setting, verifying openness typically requires explicit construction. In functional analysis, the [Open Mapping Theorem](/theorems/631) (Banach's theorem) states that a surjective bounded linear operator between [Banach spaces](/page/Banach%20Space) is automatically an open map — a deep result with no purely topological analogue. ## Open Covers and the Lindelöf Property Open sets serve as the building blocks for **covers** — collections of sets whose union contains the space. The theory of open covers connects the local structure of open sets (each point is in some open set of the cover) to global properties like compactness and paracompactness. ### Open Covers An open cover encodes a "local-to-global" problem: each point has an open neighbourhood with some property (contained in a coordinate chart, covered by an estimate, belonging to a nice set), and the question is whether finitely many of these neighbourhoods suffice. [definition:Open Cover] Let $(X, \tau)$ be a topological space. An **open cover** of $X$ is a collection $\{U_i\}_{i \in I}$ of open sets with $X = \bigcup_{i \in I} U_i$. A **subcover** is a subcollection that still covers $X$. A **finite subcover** is a subcover with finitely many elements. [/definition] Compactness — the requirement that every open cover has a finite subcover — is the strongest finiteness condition on open covers. Between "all covers have a finite subcover" (compact) and no restriction at all, there is a useful intermediate condition: every open cover has a *countable* subcover. ### The Lindelöf Property In many spaces that arise in analysis, uncountable open covers can always be reduced to countable ones. This property — named after Ernst Lindelöf — holds automatically for second-countable spaces. [definition:Lindelöf Space] A topological space $X$ is **Lindelöf** if every open cover of $X$ has a countable subcover. [/definition] Every compact space is Lindelöf (a finite subcover is in particular countable). Every second-countable space is Lindelöf: if $\{B_n\}_{n \in \mathbb{N}}$ is a countable basis and $\{U_i\}$ is an open cover, then for each $x \in X$ choose $U_{i(x)}$ containing $x$ and a basis element $B_{n(x)}$ with $x \in B_{n(x)} \subseteq U_{i(x)}$. The countably many basis elements $\{B_{n(x)}\}$ that appear suffice to cover $X$, and for each one, select a single $U_i$ containing it. This produces a countable subcover. In metrizable spaces, the Lindelöf property is equivalent to separability and second countability: [quotetheorem:545] This equivalence is specific to metrizable spaces. In general topological spaces, the three conditions can differ: there exist Lindelöf spaces that are not second countable (for instance, the Sorgenfrey line), and separable spaces that are not Lindelöf (under certain set-theoretic axioms). The Lindelöf property has practical consequences for measure theory and analysis. On a Lindelöf space, every open cover can be reduced to a countable one, which means that $\sigma$-additivity of a measure suffices to control the measure of arbitrary open sets. This is one reason why second-countable spaces are the natural setting for Borel measures. ## References - Munkres, J. R., *Topology* (2nd ed., 2000). - Willard, S., *General Topology* (1970). - Steen, L. A. and Seebach, J. A., *Counterexamples in Topology* (1978). - Kelley, J. L., *General Topology* (1955).