Definition:Abstract Geometry

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Let $P$ be a set and $L$ be a set of subsets of $P$.

Then $\struct {P, L}$ is an abstract geometry if and only if $\struct {P, L}$ satisfies the abstract geometry axioms:

\((1)\)   $:$     \(\ds \forall A, B \in P: \exists l \in L:\) \(\ds A, B \in l \)      
\((2)\)   $:$     \(\ds \forall l \in L: \exists A, B \in P:\) \(\ds A, B \in l \land A \ne B \)      


The elements of $P$ are referred to as points.


The elements of $L$ are referred to as lines.

The above axioms thus can be phrased in natural language as:

$(1): \quad$ For every two points $A, B \in P$ there is a line $l \in L$ such that $A, B \in l$
$(2): \quad$ Every line has at least two points

Also see

  • Results about abstract geometries can be found here.