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.

Please, I need some help with this exercise

Consider the space

$X=\bigcup_{n\in \mathbb{N}} \{\frac{1}{n}\}\times[0,1]\cup ([0,1]\times\{0\})\cup(\{0\}\times [0,1]),$

With the topology of subspace of $\mathbb{R}^2$. Show that $X$ is connected but not locally connected

share|improve this question
    
What have you tried so far? –  Asaf Karagila Oct 29 '12 at 18:54
    
the basics, but really I don't know how to do this –  Elmo goya Oct 29 '12 at 18:56

1 Answer 1

up vote 3 down vote accepted

Connectedness: The space is even path-connected as you can readily specify a path from $(x_1,y_1)$ to $(x_2,y_2)$ via $(x_1,0)$ and $(x_2,0)$.

If $X$ were locally connected, you would find in each neighbourhood of $(0,1)$ a connected open subset. But each such open neighbourhood contains some point $(\frac1n,1)$ and can be shown to be disconnected by considering the disjoint open subsets given by $x<\frac1n$ and $x>\frac1{n+1}$, respectively.

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.