# What is (a) geometry?

There is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it.

There is also no question what (universal) algebra is and what it's about. (Among other things, it's about algebras.)

But what is geometry and what is it about? Is there a thorough and generally agreed upon definition of a geometry (= geometric structure) comparable to the unequivocal definition of a topology?

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I've heard it argued that the "correct setting" for geometry is locally ringed spaces. I'd elaborate further, but I don't yet know enough to do that perspective justice. – Jesse Madnick Sep 6 '12 at 20:45
I don't think the question really makes sense for the following reason: both of the words topology and geometry have many meanings and you are comparing the meanings that are quite incomparable. Note, that topology is not just a certain collection of sets, but it is also a subject as such and it is also a phenomenon of ignoring the local details. Similarly with algebra. Now, geometry is again a subject, but it is certainly not a mathematical object in the same way topology (as a collection of sets) is! – Marek Sep 6 '12 at 20:47
"Geometry" = "measurement of the earth"... by etymology – GEdgar Sep 6 '12 at 21:36
Cf. Lurie's Derived Algebraic Geometry V, where he defines "geometries" in full generality. – Aaron Mazel-Gee Sep 6 '12 at 21:42
algebra is about the structure of sets, where structure means a function defined on it. – Vicfred Sep 9 '12 at 22:44

According to Klein, geometry can be viewed as the action of a group on a space, be it smooth or finite. See this. That is, a geometry on a set $X$ is a triple $(X,G,A)$, where $G$ is a group with action $A$ on $X$.
So would you subscribe to the following definition: "A geometry is a triple $\langle X, G, a \rangle$ with $X$ a set, $G$ a group and $a$ a group action $a: G \times X \rightarrow X$." That's it? – Hans Stricker Sep 6 '12 at 20:12