Let E be the Euclidean plane. Does there exist a collection C of subsets of E whose
union is E and which are all straight lines such that (1) No two distinct straight lines
belonging to C are parallel (2) Every subset of E which is a straight line not
belonging to C is parallel to exactly one straight line belonging to C and 
(3) No three pairwise distinct straight lines belonging to C are concurrent?
 The problem is trivial, of course, without condition (3). All the straight lines
belonging to C can intersect in the same point.
 If such a collection as C exists, could anybody give an example of one.
Or is some version of the Axiom of Choice needed to prove its existence?
 Finally, while still requiring conditions (1) and (2) to be fulfilled,
one could ask whether it is possible for the union of all the straight lines
belonging to C can be a closed proper subset of E-rather than E itself. But this,
I suspect, is impossible-even with the Axiom of Choice and even though we have not
imposed condition (3).



A question about the Axiom of Choice and straight lines in the Euclidean plane.
Such a set of lines can be constructed by transfinite induction of length $\mathfrak c = 2^{\aleph\_0}$.


Enumerate all possible directions and all possible points in order type $\mathfrak c$. (This uses a well-order of the reals.) 


In the i-th step, do the following two steps:


Step (1,i): If the i-th point is covered by the lines selected so far, do nothing. Otherwise, choose a line through this point which has a direction that was not used by previously selected lines, and that does not meet any intersection point of any two previous lines. 


Step (2,i): If a line with the i-th direction has already been selected, do nothing. Otherwise, choose a line in the i-th direction which does not contain meet any intersection point of lines chosen so far. 


Each of the choices above uses a well-order of the reals. Each such choice is possible because there are $\mathfrak c$ many possibilities a priori, and fewer than $\mathfrak c$ are disqualified by the requirement to avoid certain points. 


Unless I have made a mistake, this construction by transfinite induction is quite standard. If I recall correctly, Ciesielski's book "Set theory for the working mathematician" contains more examples of such constructions. ADDED: Also, [Joel's question](https://mathoverflow.net/questions/93601) mentions several similar (plus a few dissimilar) constructions, both on mathoverflow and elsewhere. 


