Recently I have started reading Oswald Veblen's A System of Axioms for Geometry. There it is written that (see page 346),
Axiom XI. If there exists an infinitude of points, there exists a certain pair of points $AC$ such that if $[\sigma]^*$ is any infinite set of segments of the line $AC$, having the property that each point which is $A$, $C$ or a point of the segment $AC$ is a point of a segment $\sigma$, then there is a finite subset $\sigma_1, \sigma_2,\ldots\sigma_n$ with the same property.
Here $[\sigma]$ denotes a set or class of elements, any one of which is denoted by $\sigma$ alone or with an index or subscript.
Now my question is,
What does this axiom mean in "simple" geometrical terms?
The $\S$$4$ of the paper (see page 347) gives some discussion of this axiom but I can't relate this axiom to the Heine-Borel Theorem.
Can someone help?