Section 24 Connected Subspaces of the Real Line

1. A linear continuum is an ordered set $L$ such that the least upper bound property holds and for any pair of elements there is another one between them.
2. A subspace of a linear continuum is connected iff it is a convex subset.
3. Any ordered set connected in the order topology is a linear continuum.
4. If $X$ is well-ordered then $X\times[0,1)$ is a linear continuum in the dictionary order.
5. Generalized Intermediate Value Theorem. If $f:X\rightarrow Y$ is a continuous function from a connected space to an ordered set in the order topology, and $f(a)<r<f(b)$ then there is $c\in X$ such that $f(c)=r$ .
6. Path connectedness: given $x,y\in X$  a path from $x$ to $y$ is a continuous function $f:[a,b]\rightarrow X$ such that $f(a)=x$ and $f(b)=y$ . $X$ is said to be path connected if any two points in $X$ are path connected.
7. Path connectedness implies the connectedness. But not vice versa.
8. The ordered square is connected but not path connected (the real line is just not enough for constructing the path).
9. The topologist’s sine curve$\{(x\times\sin(1/x))|x\in(0,1]\}\cup(\{0\}\times[-1,1])$ is another example.
10. If a connected subspace of $\mathbb{R}^2$ is open then it is path connected.
11. (???) If a set is path connected and Hausdorff in one topology then it is not path connected in any strictly finer topology.
12. Subspaces and path connectedness:
13. If a path connected subspace have a common point with any set in a collection of path connected subspaces then the union of the set with the union of the collection is path connected.
14. If a collection of path connected subspaces have a point in common then their union is path connected.
15. If a subspace is path connected then adding some of its limit points keeps it connected BUT it does not have to remain path connected.
16. For example, the topologist’s sine curve is the closure of a path connected set.
17. Continuous functions and path connectedness:
18. The image of a path connected space under a continuous function is path connected.
19. Products and path connectedness.
20. A finite product of path connected spaces is path connected.
21. An arbitrary product of path connected spaces is path connected in the product topology.
22. $\mathbb{R}^\omega$ is path connected in the product topology but is not path connected in the box or uniform topology (it is not even connected in those two).
23. The long line $L$ : $S_\Omega\times[0,1)$ in the dictionary order with its smallest element $(a_0,0)$ deleted.
24. $L$ is path connected and locally homeomorphic to $\mathbb{R}$ : $(-\infty,(a,0))$ is homeomorphic to an open interval in $\mathbb{R}$ .
25. $L$ cannot be embedded into $\mathbb{R}$ or $\mathbb{R}^n$ (it is not second-countable, see Exercise 7 of Section 30).