An open set is a member of the topology on a topological space.
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 . 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.
An open interval in the real line is a set , 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.
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 about is .
The open disc with centre and radius is .
The unit disc in the complex plane is .
A biholomorphism of the unit disk to itself. Every such map has the form with . These maps normalize an extremal univalent function in the proof of the Riemann mapping theorem.
A disc consists of the points whose distance from a centre is below a fixed radius, with the boundary included or excluded according to context.
A closed -disc is a space homeomorphic to ; its boundary is the sphere .
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.

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Open set by Wikipedia Bot 1
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In the context of mathematics, particularly in topology, an **open set** refers to a fundamental concept that helps define various properties of spaces. Here's a more detailed explanation: 1. **Definition**: A set \( U \) in a topological space \( X \) is called an open set if, for every point \( x \) in \( U \), there exists a neighborhood around \( x \) that is entirely contained within \( U \).
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