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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
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2 Answers

up vote 15 down vote accepted

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 $X$.

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In this context, a space may be taken to be a set. If extra structure is given, then we append more to the word "geometry". For example, if said space is metric, then we study Riemannian geometry, etc. –  user02138 Sep 6 '12 at 20:06
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
This answers the matter of what is a geometry. What is the branch of mathematics called geometry then? I'd say it is the study of group actions and their invariants, but I'm no geometer and I've always found that kind of confusing, so I'll leave it for others to answer. :) –  tomasz Sep 6 '12 at 20:19
@Hans: it is necessarily vague, since I claim there is no real answer to your question. It's the same as asking "what is math?". There is just too many geometrical subareas whose only common point seems to be what I have already said: they study some kind of generalization of those basic objects like circles and triangles in the plane. Whether those generalizations are varieties, manifolds, schemes, Banach spaces or whatever doesn't really matter. –  Marek Sep 6 '12 at 21:06
@Marek: you may be right. Nevertheless geometry and topology can be seen on a par, and there is algebraic topology opposed to algebraic geometry and the question may hope for a definite answer: what geometry is opposed to topology? –  Hans Stricker Sep 6 '12 at 21:12
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J.W. Cannon also gave a definition:

"A geometry is a topological space endowed with a proper path metric."

He also gave a definition of a geometric group action:

"A [group] action is geometric [on a set S] if S is a geometry and the action is isometric, cocompact and properly discontinuous."

These definitions can be appropriate to work in geometric group theory.

See: J. W. Cannon. Geometric group theory. In Handbook of Geometric Topology. Elsevier, 2002. (In particular p. 271-272.)

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