first of all I am new to this forum so please forgive any newbie formatting or formulation errors. Background:
I am trying to get into math by playfully reading some books I am interested in. I have enjoyed the German equivalent of A Level but am definitely rusty in my fundamentals.
Currently I am reading "Sleight of Mind - 75 Ingenious paradoxes" by Matt Cook. After a description of Hilbert's Grand Hotel Paradox, there is a section called "Closing open ends".
It reads as follows:
"Consider the task of bijecting x=[0,1] with y=[0,1]. The former is closed; the latter is open on the right. Begin matching each pair of equivalent elements in X and Y. We almost have a one-to-one correspondence except for the fact that 1 ∈ X has no mate. No problem. Match 1 ∈ X with 1/2 ∈ Y, "bumping" the latter's previous mate. Now match 1/2 ∈ X with 1/4 ∈ Y. Continue by matching every 1/2^{n} ∈ X with 1/2^{n+1} ∈ Y. We are left with a one-to-one correspondence between the desired open and closed sets, having "bumped" elements along an infinite sequence of repositionings, much like the guests in Hilbert's Grand Hotel."
Given that I have no prior education on set theory I have some difficulties understanding that. Furthermore I have to admit that the Wikipedia explanation about open and closed sets also eludes me.
Does anyone have further sources for understanding this problem. Or can explain it in his own words so that a layman could understand it ?
Many thanks in advance, hopefully in the not so far future I will be able to help people out in this kind of situations.
Greetings
Ceus
