What happens when you cut an infinitely long piece of string in two?
I've had to go Google that now you've asked and the answer I found is a surprisingly good example of how some concepts can't be explained in English (or words).
Taken from Quora
Let’s forget about cutting it “in half”, and cut it at whatever point you like.
Mathematically, we can model your infinitely long string with the set of Real Numbers R . Say you cut it at point a. Then, you are left with two pieces: R1 and R2, and either:
R1=(−∞,a)
R2=[a,+∞)
Or:
R1=(−∞,a]
R2=(a,+∞)
R represents the sets of numbers in the “Real Number Line”. The notation (−∞,a) represents the set of number smaller than a ( ⟹a not included), while [a,+∞) stands for the set of numbers larger or equal to a ( ⟹a included). Subsets of R on the form (a,b) are called open, those on the form [a,b] are called closed and those on either of the forms [a,b) or (a,b] are called half-open/half-closed.
You can see how for all cases R1 is not equivalent to R2 . In fact, within this model, it is impossible to cut the string in two equivalent pieces: One will always contain the cutting point a as its “last point” and the other one won’t even have such “last point”! If you want the pieces to be “equal”, you need to remove a and end up with:
R1=(−∞,a)
R2=(a,+∞)
Note R1+R2≠R (i.e. the pieces added together are no longer the original string, since point a is missing). But they are now equivalent by topological standards.
Now the above is all gobbledygook to me but you can see the person who's answered it has taken the time to fully explain their answer. The problem is if, like me, you don't understand the maths in the answer you won't understand the answer at all.