Open Set

An open set is a subset of a topological space that belongs to its specified collection of open subsets. In spaces equipped with a distance, a set is open when every point in it has a sufficiently small surrounding ball contained entirely within the set. This generalizes the behavior of open intervals on the real line. Open sets are fundamental to topology: they provide the language for defining continuity, neighborhoods, and compactness without requiring a numerical notion of distance. Openness depends on the topology and ambient space, rather than on a subset alone. (math.toronto.edu)

Definition in a topological space

A topology on a set $X$ is a collection $\tau$ of subsets of $X$, satisfying three axioms:

1. The empty set $\varnothing$ and $X$ belong to $\tau$.
2. The union of any family of members of $\tau$ belongs to $\tau$.
3. The intersection of any finite family of members of $\tau$ belongs to $\tau$.

The pair $(X,\tau)$ is a topological space, and $U\subseteq X$ is open precisely when $U\in\tau$. “Arbitrary union” includes unions indexed by infinite sets; the intersection requirement, by contrast, is restricted to finite families. (math.toronto.edu)

A basis describes a topology through a collection of basic open sets whose unions give all open sets. Equivalently, $U$ is open if every $x\in U$ belongs to a basic open set contained in $U$. On the real line with its usual topology, open intervals form a basis. A basis need not contain every open set, but it determines the entire topology. (math.toronto.edu)

Metric characterization

In a metric space $(X,d)$, the open ball centered at $x$ with radius $r>0$ is

A subset $U\subseteq X$ is open in the metric topology if

The radius may depend on the point; no single radius must work throughout the set. Every open ball is open, as follows from the triangle inequality. Consequently, open sets are exactly unions of open balls. (jirilebl.github.io)

For the real numbers with distance $d(x,y)=|x-y|$, the interval $(a,b)$, where $a<b$, is open: at any $x\in(a,b)$, choose $r<\min(x-a,b-x)$. The interval $[a,b)$ is not open in $\mathbb R$, because every ball centered at $a$ contains points less than $a$. Open rays such as $(a,\infty)$ are also open, so openness does not imply boundedness. (jirka.org)

In $\mathbb R^n$, the same definition uses Euclidean distance. Open disks in the plane and higher-dimensional open balls provide the corresponding local surrounding regions. (jirilebl.github.io)

Set operations and related concepts

Infinite intersections of open sets need not be open. For example, in the usual real topology,

and the singleton $\{0\}$ contains no open ball around its point. Finite intersections and arbitrary unions, however, always remain open. (jirka.org)

A closed set is a subset whose complement is open. Open and closed are not mutually exclusive classifications: the empty set and whole space are both. A set having both properties is called a clopen set. A subset may also be neither open nor closed, as $[0,1)$ is in the usual real topology. (jirka.org)

A neighborhood of $x$ is commonly defined as a set containing an open set that contains $x$; under this convention, a neighborhood need not itself be open. The interior of $A$, denoted $\operatorname{Int}(A)$, is the union of all open subsets contained in $A$. Thus $A$ is open exactly when $A=\operatorname{Int}(A)$. In a metric space, this means every point of $A$ has a ball lying inside $A$. (jirilebl.github.io)

Relative openness

For $Y\subseteq X$, the subspace topology consists of the sets

where $V$ is open in $X$. Such a set is described as open in $Y$ or relatively open. It need not be open in the larger space. (math.utoronto.ca)

For example, $[0,\tfrac12)$ is open in $Y=[0,1]$, because it equals $Y\cap(-1,\tfrac12)$, but it is not open in $\mathbb R$. Likewise, a circle with its topology inherited from the plane has relatively open arcs, although these arcs are not open subsets of the plane. Basic open sets for a subspace are obtained by intersecting the subspace with basic open sets of the ambient space. (math.utoronto.ca)

Continuity, compactness, and measurability

A function $f:X\to Y$ is a continuous function precisely when $f^{-1}(V)$ is open in $X$ for every open $V\subseteq Y$. A homeomorphism is a bijection whose map and inverse are continuous; it therefore carries open sets to open sets in both directions. (math.toronto.edu)

An open cover is a family of open sets whose union contains the space under consideration. A compact space is one for which every open cover admits a finite subcover. These definitions use open sets rather than distances, so they apply to general topological spaces. (math.toronto.edu)

In measure theory, open sets generate the Borel sigma-algebra, the smallest sigma-algebra containing them. Every open set is therefore Borel measurable, while Borel sets also include closed sets and sets obtained through countable unions, countable intersections, and complements. (math.toronto.edu)