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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?

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  • $\begingroup$ Search for veblen axiom xi in booksgoogle.com . $\endgroup$ – Tony Piccolo Dec 24 '15 at 14:44
  • $\begingroup$ Axiom XI is a continuity axiom. In general continuity axioms are axioms that are needed so that lines (or other figures) that look to intersect are also provable to intersect. and in the other direction lines that do not intersect also cannot look to intersect (maybe this last idea is more important) Without a continuity axiom this is sometimes not possible to prove, or disprove. See also pages 368-371 of Velben's article ( theorems 33-43) Good luck $\endgroup$ – Willemien Feb 26 '16 at 14:31

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