# Definition:Abstract Geometry

## Definition

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

Then $\left({P, L}\right)$ is an abstract geometry if and only if:

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

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 geometry can be found here.