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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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    the basics, but really I don't know how to do this2012-10-29

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enter image description hereConnectedness: 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.

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    Am I correct? Here there is no way of uploading the picture. please verify my drawing.2018-11-14
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    I will surely delete the picture. sorry for altering the answer.2018-11-14
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    By considering $(X,\mathscr T_{X})$ as the subspace topology of usual topology of $\mathbb R^2$2018-11-14
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    I got the separation of the slice in the open ball(say B). $U=B\cap \{(t,y)\mathbb R^2:t and $V=B\cap \{(t,y)\mathbb R^2:t>x\}\cap X$ are the separation right? so it is not locally connected.2018-11-14
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    Am I correct?I request you to verify.2018-11-14