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More Information on Why "Infinity" Does Exist in the Context of A Topological Space

Remember, we are trying to see why the answer is "yes" to the following question:

Does there exist some topological space (that is, a set of objects plus a definition of what convergence means) which, as well as including the familiar real numbers we are used to, also includes an "infinity" concept to which some sequences of real numbers converge?

• Start with the real numbers, take any additional object you like, add it to the set of real numbers, and make the resulting set into a topological space. It turns out that all you need to do to make a set into a topological space is come up with an appropriate collection of subsets (which in the case of the real numbers are unions of open intervals). It would take us too far afield to discuss exactly why specifying an "appropriate collection of subsets" is all it takes to completely determine all notions of which sequences converge and which don't. However, that is in fact all it takes, and in our new set (the real numbers together with the additional object) there does turn out to be an appropriate collection of subsets which makes it into a topological space, and the resulting notion of "convergence" is such that sequences like 1, 2, 3, ...converge to the additional object, which justifies calling the additional object "infinity".

• If the previous construction seems too abstract, think about drawing the real numbers on an interval of finite length, instead of the usual infinitely long number line. (It may seem surprising that this can be done, but all you need to do is compress the distances so that larger numbers are drawn closer together. For instance, you could draw the number 1 at a distance 1 from the end, 2 at a distance 1/2 from the end, 3 at a distance 1/3 from the end, and so on. The following picture illustrates this (not quite to scale):

          *********************************************** <-- the interval
             ... -1          0          1     2   3 4 ...
           
  

  Now you can take the endpoints of the interval, call them "infinity" and "-infinity", and say that a sequence of numbers a_1, a_2, a_3, ... "converges to infinity" if and only if the dots where those numbers are drawn converge to the right-hand endpoint of the interval.

No matter which approach you take, you can define a topological space which has the real numbers as a subset, also has an additional object called "infinity", and in which the notion of convergence is such that sequences like 1, 2, 3, ... converge to this additional object. In this sense, therefore, infinity exists.