# Connectedness of the comb space

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

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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
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.