We define a closed ray in a topological space $X$ to be its closed subset homeomorphic to the real half-line $[0,\infty)\subseteq \mathbb{R}$. Call a topological space $X$ rayless if it does not contain a closed ray.

By a theorem of Okhezin a contractible polyhedron (i.e. geometric realization of a simplicial complex) has the fixed point property if and only if it is rayless (see this answer). Besides Okhezin's work in fixed point theory I do not remember seeing non-compact rayless polyhedra (or more general rayless topological spaces) in the literature, except for those serving as simple counterexamples.

On the other hand, there is some research concerning rayless graphs, which are graphs not containing an infinite simple path (just do a Google or MathSciNet search for "rayless graph" to find some of it). I also have some experience with rayless posets and find them to have nice properties, often similar to those of finite posets; some of them also transfer to rayless simplicial complexes (i.e. those with rayless geometric realization, or equivalently, with rayless $1$-skeleton).

I wonder whether the class of rayless polyhedra has been investigated, possibly under some different name? Do they appear in mathematics in some natural way? Is there work on rayless polyhedra (published or not) that I am not aware of?

EDIT (some motivation for the rayless conditon): Note that there are two ways in which a connected polyhedron can fail to be compact: either it contains a closed ray or it is not locally compact. Therefore, the classes of locally compact polyhedra and rayless polyhedra are both quite natural generalizations of the class of compact polyhedra, and those generalizations are "orthogonal": a connected polyhedron that is both rayless and locally compact must be compact.


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