Open set ($U\in \tau$)

An open set is a member of the topology on a topological space. In a metric space, a subset $U$ is an open set precisely when every $x\in U$ has an open ball $B(x,r)$ contained in $U$ for some $r>0$. For $U=(0,1)$, one can take $r=\frac{1}{2}\min (x,1-x)$. At $x=0.2$, this gives $r=0.1$ and $B(x,r)=(0.1,0.3)\subset U$. The radius depends on the point; no uniform radius is required. ^[ref]

Openness depends on the ambient topological space. In $A=[0,1]$ with its subspace topology, $[0,1/2)=A\cap (-1,1/2)$ is an open set in $A$, although it is not an open set in the real line. Similarly, the whole closed interval $A$ is both an open set and a closed set in itself. These relative open sets are what the continuity test uses for maps involving a subset with its subspace topology.

Arbitrary intersections of open sets need not be open sets: ${\bigcap }_{n\ge 1}(-1/n,1/n)=\{0\}$, a singleton that is not an open set in the usual real line. Any point $x\ne 0$ is excluded once $n>1/|x|$. This is why the topology axiom specifies finite intersections.

A domain is a nonempty connected open set in the ambient space under discussion. In complex analysis this normally means an open connected subset of $\mathbb{C}$. A star-shaped set that is open is a domain, but a domain need not be a simply connected domain and can have a period obstruction to a holomorphic antiderivative.

A clopen set is a subset that is both open and closed.

This is the open-boundary case of an interval.

An open interval in the real line is a set $(a,b)=\{x:a<x<b\}$, allowing either endpoint to be infinite.

A nondegenerate interval contains more than one point; for an interval with finite endpoints, this means that its endpoints are distinct.

A neighbourhood of a point is a set containing an open set that contains that point.

An open neighbourhood of a point $x$ in a topological space $(X,\mathcal{T})$ is an open set $U$ containing $x$. A general neighbourhood need only contain an open neighbourhood; it need not itself be open. A finite intersection of open neighbourhoods of the same point is again an open neighbourhood, by the topology axioms.

The neighbourhood system at a point consists of all its neighbourhoods. A neighbourhood basis is a subfamily cofinal under reverse inclusion.

A neighbourhood basis at a point is a family of neighbourhoods such that every neighbourhood of the point contains one of them.

In a metric space, the open ball of radius $r>0$ about $x$ is $\{y:d(x,y)<r\}$; it is the open-boundary convention for a ball.

A punctured ball is an open ball with its centre removed. In the real line it has two connected components. In Euclidean space of dimension $n\ge 2$ it is path-connected: for any two points, choose a third point inside the ball outside the two lines through the centre and those points. Such a point exists since two lines do not cover an open ball in dimension at least two. The two line segments joining through that point stay inside the ball, which is a convex set, and avoid its centre. They form a continuous path.

An annulus is the region between two concentric circles.

Rotation through any nonzero angle defines a continuous self-map of a circular annulus with no fixed point. Hence an annulus does not have the fixed-point property.