I am trying to understand the concept of homotopy. And I am wondering what are examples of "purely" homotopical properties?
I read from a book about Homotopy Type Theory, that
..., in homotopy type theory we think of it [a : A] instead as: "a is a point of the space A".
We should stress that these “spaces” are treated purely homotopically, not topologically. For instance, there is no notion of “open subset” of a type or ... We only have "homotopical" notions, such as paths between points and homotopies between paths, which also make sense in other models of homotopy theory (such as simplicial sets).
However, when I read about homotopy in Wikipedia, the description seems to be all based on topology. For example, homotopy is defined as a continuous deformation $H: X \times [0,1] \to Y$ from one continuous map $f: X \to Y$ to another $g: X \to Y$. But if we don't have notions of open sets, how do we have the concept of continuous maps and their deformations in the first pace?
From a different perspective, I know that there are properties that are geometric but not topological (e.g. two lines being parallel etc.), and there are concrete examples of topological properties, e.g., that two lines are connected/intersect each other/are disjoint etc.
What are the notions that are purely homotopical?
BTW, I don't know much about topology/homotopy.
The notion of a topology is needed not for defining homotopy itself but rather for defining the notion of a continuous map, without which you can't even start doing Topology (in the sense of the field within mathematics). To get an idea of what kind of property is preserved under homotopy equivalence, it is a good idea to do lots of exercises on the fundamental group. Once you are comfortable with that, go on to homotopy groups. In the context of CW complexes, spaces will be homotopy equivalent if and only if there is a map between them inducing isomorphism on all homotopy groups. In that sense, homotopy groups give you precisely all the information on the homotopy type of a space.