Say that a topological space $X$ is **locally connected at some point $x$**, if it has a local base at that point consisting of connected open sets.
Also $X$ is **locally connected** if it is locally connected at each of its points.

It is well known that a connected space is not necessarily locally connected.

For example, let $X = \bigl([0;1] \times (\mathbb{Q}\cap[0;1])\bigr) \cup (0\times [0;1])$ be the union of horizontal segmets with rational $y$-coordinate together with a vertical segment intersecting all of them. Then $X$ is connected but not locally connected. Namely it is not locally connected at any point except for the points of that vertical segment $0\times[0;1]$. On the other hand, it is still locally connected at each point of $0\times[0;1]$.

In fact, in all such examples which I know, connected spaces are still locally connected at some of their points.

The question is to

construct an example of a connected space being not locally connected at any of its point.

Probably this is also a standard example, but I never meet it. I would be grateful for any information about that question.

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