If I want to consider the family $\{U_{a}\}_{a\in J}$ of path-connected subsets of a topological space $X$, can I assume they are pairwise disjoint? This would seem intuitively true. I need to prove that the homotopy classes $[\{-1,1\},\{1\};X,\{x\}]$ are in bijective correspondance with the $U_{a}$'s. I think I have an idea if I can assume that the $U_{a}$'s are disjoint!
Decomposing a space into path connected components
0
$\begingroup$
general-topology
-
0In the statement of the problem it says components. I didn't read into the terminology too much. I should have looked up the term. :S Thanks again! – 2012-02-02
1 Answers
1
Yes, two path components are disjoint. The key point is that the union of two path-connected sets with a common point (say, $p$) is also path-connected. Indeed, one can go from any point of one set to $p$ and from there to any point of another set.
So, if two path components had a common point, their union would be a strictly larger path-connected set, which is a contradiction.