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Definition of Infinite Broom

In wikipidea it is written that

The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point $(1, 1/n)$ as n varies over all positive integers, together with the interval $(1/2, 1]$ on the x-axis

Diagram of infinite broom

enter image description here

My thinking : From the diagram ,we conclude that all the line segment start from $(0,0)$ to $(1,1/n)$ and there is no any line segment start from $(1/2,1]$

I don't understand this sentence together with the interval $(1/2,1$] on the $x$ axis

why together with the interval $(1/2, 1]$ on the x-axis ?

jasmine
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    The infinite broom is an example similar to that of the topologist's sine curve which is connected but not path connected. The interval $\left(\frac12,1\right]$ ensures that it isn't path connected. – Rushabh Mehta May 16 '21 at 14:38

1 Answers1

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The extra $(\frac12,1] \times \{0\}$ makes it a better counterexample to some common ideas: it's connected but not path-connected and not locally (path-)connected at the $x$-axis points either. It's its raison d'être, as it were. It's similar to the topologist's sine curve (see here, e.g.) in that respect.

Henno Brandsma
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