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M W cleverly answered my question at https://mathoverflow.net/questions/454997/ asking for a space without unique sequential limits, but with closed retracts, using this example:

Let $X=[0,\infty)\cup \{\infty_1,\infty_2\}.$ The topology on $[0,\infty)$ is the usual Euclidean topology, a neighborhood base of $\infty_1$ is given by sets of the form $\{\infty_1\}\cup (a,\infty)\backslash 2\mathbb N$, and a base for $\infty_2$ by sets of the form $\{\infty_2\}\cup (a,\infty)\backslash (2\mathbb N +1)$.

I'm making a contribution to pi-Base to give a result for this search, and would like to include properties on its connectedness. For example, it is certainly connected:

Consider a clopen subset $C$ of the space containing $\infty_1$. Note $\infty_2$ is in the closure of any neighborhood of $\infty_1$, so $\infty_2\in C$. Then, any proper open neighborhood of both $\infty_1$ and $\infty_2$ is of the form $(a,\infty)\cup\{\infty_1,\infty_2\}$; but this is not closed, so we conclude $C=X$.

But what about path connected? Strongly connected? Biconnected?

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    $\begingroup$ For connected, $[0,\infty)$ is connected, and dense. Hence $X$ is connected as its closure. $\endgroup$
    – PatrickR
    Commented Dec 15, 2023 at 22:55
  • $\begingroup$ Much slicker than mine, thanks. $\endgroup$ Commented Dec 15, 2023 at 23:04

1 Answer 1

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  1. $X$ is not path connected:

Suppose there is a path $\gamma:[0,1]\to X$ joining $\infty_1$ to some $b\in[0,\infty)$. By collapsing part of the path if necessary, we can assume $\gamma$ only takes the value $\infty_1$ at $t=0$.

Take a nbhd of $\infty_1$ of the form $V=\{\infty_1\}\cup (a,\infty)\backslash 2\mathbb N$ and $t>0$ such that $\gamma$ maps $[0,t]$ into $V$. So the restriction of $\gamma$ to $[0,t]$ is a path in $V$ from $\infty_1$ to some $b<\infty$. But the connected components of $V$ are all open intervals of the form $(2k,2k+2)$, together with the singleton $\{\infty_1\}$. So there is no possible path in $V$ from $\infty_1$ to $b$. And similarly at $\infty_2$.

The path connected components of $X$ are $[0,\infty)$, $\{\infty_1\}$ and $\{\infty_2\}$.

  1. $X$ is not locally connected:

A nbhd of $\infty_1$ of the form $V=\{\infty_1\}\cup (a,\infty)\backslash 2\mathbb N$ does not contain any connected nbhd of $\infty_1$, by the same consideration of the connected components of $V$ as above. So $X$ is not locally connected at $\infty_1$. And similarly at $\infty_2$.

(And hence $X$ is not locally path connected or locally arc connected at these two points.)

  1. $X$ is not strongly connected:

The map $f:X\to[0,1]$ sending any point of $[0,1]\subseteq X$ to itself and any other point to $1$ is continuous and non-constant.

  1. $X$ is not biconnected:

$X$ can be partitioned into two connected sets, each with at least two points. For example, the set $[0,1)\subseteq X$ and its complement (which is homeomorphic to $X$, hence connected).

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