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Let $X$ be a topological space and let $\left\{ Y_{i}\right\} _{i=1}^{\infty}$

be a descending chain of closed connected subsets of $X$. I know from reading elsewhere that ${\displaystyle \bigcap_{i=1}^{\infty}Y_{i}}$ is not necessarily a connected subspace of $X$ but I have no counter example and I haven’t managed to come up with one.

There is a counter example here to the same question while also assuming $X$ is compact. However, it uses the quotient topology which I haven’t studied about so I would prefer a different counterexample that does not use the quotient topology.

Help would be appreciated!

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The answer here can also be said without the quotient:

Let $X:=[-1,1]\cup\{0’\}$ where $0’$ is a new, fictive element, playing the role of a *second origin*, and define the topology on $[-1,1]$ as usual, and let $(-a,b)\setminus\{0\}\cup\{0’\}$ be a base for open neighborhoods of $0’$.

Now consider $Y_n:=[-1/n,1/n]\cup\{0’\}$.

Try the following sets:

$$Y_i = \bigg([-2,2]\times\mathbb{R}\bigg) – \bigg((-1,1)\times(-i,i)\bigg).$$

That is, $Y_i$ is a closed infinite strip with successively larger open boxes removed. The intersection

$$\bigcap_{i=1}^\infty Y_i = ([-2,-1]\cup[1,2])\times\mathbb{R}$$

is disconnected, but each $Y_i$ is closed and connected.

In $\mathbb{R}^{2}$ take two disjoint infinite lines $L_{1},L_{2}$ connected by vertical lines $B_{1},…$ (like a ladder). Then define

$$A_{n}=L_{1}\cup L_{2}\cup \bigcup_{k\geq n}B_{k}.$$

These form a descending chain and their intersection will be $L_{1}\cup L_{2}$.

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