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Today in class my professor said that the sets $[0,1]$ and $[0,2]$ have the same amount of points. I have some questions about this. Since $[0,2] = [0,1] \cup [1,2]$, doesn't it guarantee $[0,2]$ to have more points that $[0,1]$? Also, how exactly can we show $[0,2]$ has as much points as $[0,1]$? Any explanation or keywords that I should Google is appreciated.

Edit: I noticed that the different topology thing is referring to the real number line $[0,1]$ and square with length $1$. Sorry for the confusion.

Alexis Olson
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Harry
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  • Are you sure you copied it correctly? $[0,1]$ and $[0,2]$ are homeomorphic. – carmichael561 Oct 12 '16 at 18:43
  • In a class there is the same number of students and chairs if they sit down and every student has a chair and in each chair there is a student. That is, there is a bijection between students and chairs. Isn't it $x\in [0,1]\to 2x\in [0,2]$ a bijection? This is the reason we say they have the same amount of points. On the other hand I have no clue about what your teacher said. But both sets are homeomorphic and so they have the same topology. – mfl Oct 12 '16 at 18:46
  • Depends on what you mean by "the same topology." Perhaps he was comparing $[0,1]$ and $[0,2)$ or $(0,2)$? If you haven't covered what is meant for two infinite sets to be the "same size" yet, then you should just figure it is something that you will learn later. – Thomas Andrews Oct 12 '16 at 18:46
  • To be more specific, he said the real number line from 0 to 1 and 0 to 2, I thought they were the same. He also mentioned something about finding a bijective map between the 2. – Harry Oct 12 '16 at 18:46
  • @Harry This is an important subtlety when dealing with infinite sets. Set A can have more elements than Set B (in the sense of inclusion) even though the number of elements is the same. It's not at all intuitively obvious how to count the number of elements of an infinite set: someone had to think of a way to do it and justify to others that it is a useful definition. – Erick Wong Oct 12 '16 at 22:59

2 Answers2

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There are two questions here:

  • How do $[0, 1]$ and $[0, 2]$ have the same number of points?

and

  • How do $[0, 1]$ and $[0, 2]$ have different topologies?

The second question is in fact incorrect: $[0, 1]$ and $[0, 2]$ are topologically the same. So I assume something was incorrectly copied here.

The first question, however, makes perfect sense, so let me address it. When we're comparing the number of elements of two infinite sets, we do so using cardinality: two sets have the same cardinality if there is a bijection between them (that is, a way to "match up" elements of one set with elements of the other set). Note that even though $[0, 1]\subsetneq[0, 2]$, there is a bijection between them, given by $f(x)=2x$. Similarly, there are as many even numbers, or square numbers, or prime numbers, as there are integers, and as many integers as there are rationals.

See this other question for more details, including some motivation as to why this is a reasonable way to compare infinite sets in the first place.

Community
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Noah Schweber
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To your questions:

(1) The number of points is the same because there is a bijective function between them (in fact there are many) but one such is $f: [0,1]\to[0,2], x\mapsto 2x$>

(2) The second question is a bit bizarre as $[0,1]$ and $[0,2]$ are homeomorphic, but perhaps on a more pedantic level he meant the topologies are not literally equal, which is obvious as $[0,1]$ is not an open subset of $[0,2]$ in the subspace topology inherited from $\Bbb R$.

Adam Hughes
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