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Take the definition for a locally connected space:

A space is locally connected at $c$ if for every neighborhood of $c$ contains an connected open neighborhood of $c$. A space is loccally connected if it's locally connected at all points.

Now what if we modify the definition to use punctured neighborhood of $c$, that is a set $P$ such that $P\cup \{c\}$ is open. That is:

A space is ? at $c$ if for every neighborhood of $c$ contains an open neighborhood $U$ of $c$ such that $U\setminus \{c\}$ is connected. A space is ? if it's ? at all points.

Is there a name for such spaces?

Example of such a space is $\mathbb R^2$, given $c$ and a neighborhood $N$ of $c$ we will have an $\epsilon>0$ such that $U=B_\epsilon(c)\subset N$ and we have that $B_\epsilon(c)\setminus\{c\}$ is connected.

On the other hand $\mathbb R$ is not such a space since given $c$ and a neighborhood then ther must be an $\epsilon$ such that $(c-\epsilon,c+\epsilon)\subset U$ and $(c-\epsilon,c+\epsilon)\setminus\{c\}$ is not connected.

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  • $\begingroup$ By your definition, the real line is not locally connected but the plain is. $\endgroup$ Jul 5, 2017 at 8:12
  • $\begingroup$ @WilliamElliot Note that this is not locall connectedness, it's something else which I don't know the name of. My examples show that the real line is not such a space, but the plane is. $\endgroup$
    – skyking
    Jul 5, 2017 at 8:28
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    $\begingroup$ It's a bit like having no local cut points. $\endgroup$ Jul 5, 2017 at 9:09

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