Definition:Abstract Geometry
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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): $\S 2.1$