Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
4  
Do you want path components‌​? Otherwise, you can say something silly like, "I have $[0, 1]$, and $[0, 1)$ and are $(0, 1]$ are distinct path-connected subsets that overlap." –  Dylan Moreland Feb 2 '12 at 20:18
    
In 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! –  Kyle Schlitt Feb 2 '12 at 20:22

1 Answer 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.