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I'm having trouble seeing whether this space is connected or locally connected. $$X = \bigcup_{n \in \mathbb{N}} \{ x^2 + y^2 = 4 + \frac{1}{n}\} \cup\{ (x-1)^2+y^2=1\} $$

I would say there is a separation but i'm not sure, it's too straightforward to be true. So my "guess" is that one set can be the first big union and the other set what is left. And for local connectedness I believe the point $(2,0)$ is the one that doesn't have a basis of connected sets.

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    The first big union isn't closed in $X$, so you can't use that as one of the parts of a separation. You must split the union in a separation. You're right about the point $(2,0)$. – 2017-02-01
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    @DanielFischer so what is the separation? If there is one... – 2017-02-01
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    There are lots of separations. What can you say about the circle $\{ x^2 + y^2 = 5\}$? – 2017-02-01
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    Oh, sure. One must also rest. Thanks – 2017-02-01

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