Is there any point-set definition of simple connectedness?

Question

The definition of path-connectedness refers to the set of real numbers, $\mathbb{R}$. (More precisely, the interval $[0,1]$) On the other hand, connectedness is defined "purely" in terms of points and sets, and does not refer to $\mathbb{R}$.

The definition of simple connectedness also heavily depends on the interval $[0,1]$, or $S^1$. So here's my question: Is there any notion which is very similar to the usual simple connectedness, but defined "purely" in terms of points and sets? If there is one, is this useful?

Answer 1

For many spaces, such as CW complexes, path connectedness is equivalent to connectedness. Also, the loop space of a CW complex has the homotopy type of a CW complex (this theorem of Milnor may need some extra assumptions, like finiteness, I don't know the exact statement). Thus, for CW complexes at least, simple connectedness can be defined in terms of connectedness alone : a CW complex is simply connected iff it is connected and its loop space (at some (any) point) is connected. The same then holds for higher connectivity (unless there is some finiteness hypothesis in Milnor's theorem.)

Answer 2

This is an extended version of my comment. There are two competing homotopy theories which address the question that you asked, these are Čech and Steenrod (or Steenrod-Sitnikov) homotopy theories. You can find a detailed discussion of these for instance here. However, if you are just learning general topology and algebraic topology, both theories will be very hard to swallow. A slightly more elementary theory is the one of Čech homology (or cohomology) groups, see here. The idea is to "approximate" the given (pointed in the homotopy case) topological space $X$ with a sequence of simplicial complexes $C_i$, which are nerves of certain open covers of $X$. For each $C_i$ one can define (co)homology and homotopy groups purely combinatorially. (These groups form a natural inverse system and one then can take the inverse limit. This is Čech's construction. The one due to Steenrod is even more complex.) You might be familiar with this construction in the context of homology groups. Note that the notion of a simplicial complex, while motivated by the geometric notion of a simplex (which does require real numbers) is in fact purely combinatorial, like the more elementary notion of a graph. Hence, its (co)homology groups can be defined combinatorially. For instance, to define the combinatorial $\pi_1$, you can think of the elements of this group as equivalence classes of based edge-paths. This equivalence is generated via elementary homotopy of such edge-paths, where an edge of a 2-simplex $s$ is replaced by the concatenation of the two other edges of $s$ (and vice-versa).

In the case of "nice" topological spaces (like CW-complexes), the homotopy and (co)homology groups thus defined are isomorphic to the ones defined via maps of intervals/cubes/spheres and singular/cellular simplices.

Needless to say, I suggest you first get comfortable with the standard algebraic topology (e.g. in Hatcher's book) before trying to deal with anything like Čech homotopy groups.

Answer 3

The notion of covering is purely "points and sets": a map $\pi:Y\to X$ is a covering if it is surjective and every $p\in X$ has an open neighborhood $U\subset X$ such that $\pi^{-1}(U)$ is a disjoint union of copies of $U$. Now, let $X$ be a connected topological space. You can say that $X$ is simply connected if it has no nontrivial connected covering. This coincides with the usual notion of simply connectedness when $X$ is reasonable.