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While learning Topology from S. G Krantz I have a question in proof of a set contructed with the help of Cantor Set.

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Question is in last line of image.

I am unable to understand why K/{p} will be totally disconnected.

Why should K/{p} doesn't any connected component which is not singleton?

I see no reason.

  • Any connected subset has to lie within one of the lines $L_x$ (some line through $p$ would form a separation otherwise). But each $L_x^*/{p}$ is totally disconnected. It is not at all plain to me that this space is connected. – David Mitra Dec 02 '20 at 16:43
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    I do not think the $L_x^$ are defined correctly. It should be (if this is "Cantor's Teepee"): if $x\in Q$, then $L_x^$ is the set of all points in $L_x$ with rational second coordinate. If $x\in P$, then $L_x^*$ is the set of all points in $L_x$ with irrational second coordinate. – David Mitra Dec 02 '20 at 17:03

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