Understanding Erlangen Program I am trying to understand the idea of Erlangen Program. Roughly I have understood that it says geometry is the study of invariants under certain transformations.
But the following is not clear to me:
Suppose we have a set $X$ and transformation set $T$. Then we have to study about the subsets $Y$ of $X$ satisfying $g(Y)=Y$ for all $g\in T$ or we have to study some properties of $X$ which are invariant under transformations?
Here is a reference: https://encyclopediaofmath.org/wiki/Erlangen_program
 A: The idea of Klein's Erlangen program is to classify geometries according to different groups of transformations. For example, if you take the group of isometries on the plane, you get Euclidean geometry. In that geometry, the objects of study are those that preserve their nature under rigid motions. Examples are: a triangle, a circle, a straight line, etc. But if you increase the group including, say all affine transformations that can deform shapes, something like a circle is not a proper object of this geometry anymore, because it becomes an ellipse under dilation. But an ellipse cannot become a hyperbola, so hyperbolas and ellipses are different shapes in this geometry. The concept of parallelism, however, remains meaningful. You can extend your group further to include projective transformations, which are capable of transforming any non-degenerate conic into any other, so the distinction between ellipses and hyperbolas completely disappear. Parallelism is also meaningless since parallel lines meet at a point just like any other pair of straight lines (the concept of straight line is meaningful). An extreme example is topology, where any bicontinous transformation is allowed. In such a fluid world, no particular shape makes sense, but an open curve is different from a closed one.
