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Let $X$ be a compact, connected, locally connected space. Let $U$ be a connected open subset of $X$. Let $p\in \overline U$. Clearly $U\cup \{p\}$ is connected.

Is $U\cup \{p\}$ locally connected?

Is $U\cup \{p\}$ path-connected? (i.e. is there an arc from $p$ into $U$?)

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    $\begingroup$ While the answer is negative in general, it becomes positive if you assume that $X$ is a metrizable Peano continuum and $U$ is uniformly locally connected (meaning that for every $\varepsilon>0$ there is $\delta>0$ such that if $d(x,y)<\delta$, then there is a connected set $C$ containing $x$ and $y$ with diameter smaller than $\varepsilon$). $\endgroup$ Commented Nov 15, 2023 at 17:08

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Answer to both questions is "no". Take

$$X=[0,1]^2$$ $$U_n=\bigg(\frac{1}{n}-\epsilon_n,\frac{1}{n}+\epsilon_n\bigg)\times[0,1]$$

We chose each $\epsilon_n$ to be small enough so that $U_n$ are pairwise disjoint, e.g. $\epsilon_n=\frac{1}{3n}$. We then define

$$V=(0,1)\times[0,\frac{1}{2})$$ $$U=V\cup\bigcup_{n=0}^\infty U_n$$

In other words this is a "fat" comb space. Fat to make it open, but it still has similar properties: every point in $\{0\}\times[0,1]$ belongs to its closure. However both answers are "no" if $p\in\{0\}\times(\frac{1}{2},1]$.

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