nLab star domain

Idea

A star-shaped domain is a subset $U$ of an appropriate ambient space for which there exists a basepoint that can be connected to every other point by a “straight line in $U$”. Such a subset is a contractible space, using the contraction that travels each point back to the basepoint along the given lines.

Definition

For $X$ a real affine space, a star domain about a point $x\in X$ is an inhabited subset $U \subset X$ such that with $y \in X$, the straight line segment connecting $x$ with $y$ in $X$ is also contained in $U$.

For $V$ a vector space, a star domain about the origin is an inhabited subset $U \subset V$ that is radial.

These definitions can be modified in various obvious ways. For example, a star shaped neighbourhood of a point $x$ in an affine space $X$ is an open neighbourhood $U \subset X$ of $x$ that is a star domain about $x$. Or, a subset is a star domain if it is a star domain about one of its points.

A useful special case pertains to a simplicial complex $K$, where if $v$ is a vertex of $K$, then the open star of $v$ is the union of the interiors in ${|K|}$ of all the simplices containing $v$. Open stars of vertices provide a good open cover of a simplicial complex.

A convex set is the same as a set that is a star domain about each of its points.

Related entries

• open ball

• good open cover