Definition:Abstract Geometry

From ProofWiki
Jump to navigation Jump to search


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.