Definition:Abstract Geometry

Definition

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:

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

Points

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

Lines

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.

Sources

• 1991: Richard S. Millman and George D. Parker: Geometry: A Metric Approach with Models (2nd ed.) ... (next): $\S 2.1$