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Show that the Cut plane $ X={\mathbb{C}} {\setminus} ({-\infty},0] $ is polygonally path-connected by using a directed line segment in the form of ${\gamma}(t)=tz+(1-t)w $ for some interval $[w,z]$ and $0

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    Can you please explain where you're stuck? (E.g., do you not understand the definition of "polygonally path-connected", do you not know what your set $X$ looks like, can you not see intuitively why $X$ is polygonally path-connected, do you not see how to use the hint, ...?) – 2017-01-23
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    Can you see how for any point $p\in X$ the straight line connecting $p$ to $1$ lies in $X$? How does this imply the result? – 2017-01-23
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    @AndrewD.Hwang I can see how any two points are connected and I see how the cut plane will definitely be path-connected, I'm just unsure how to formally write this down. – 2017-01-23
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    How is this related to functional analysis? – 2017-01-23
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    What AndrewD.Hwang means is that if you want to connect $a$ to $b$, a general solution uses **two** such segments, [a,1] and [1,b] for example. – 2017-01-23
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    @EShawn: Does s.harp's hint help...? :) If not, sketch the cut plane, draw the point $1$, use your directed line segment formula to show that if $p \in X$, then the image of the segment from $p$ to $1$ lies in $X$ (this is the only step requiring "real work"). Can you conclude from there? – 2017-01-23

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