# Connectedness of a certain space with closed retracts but non-unique sequential limits

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?

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

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).