I don't see how this is poor UX. If we accept your statement that the terms "open" and "closed" are too similar, what do you suggest?
Also, any notation will have to be learned. Concise ones a bit more, but there is nothing one can do about that.
And that perfectly clear notation requires you to learn quite a bit by memorization, such as the meaning of those {} brackets, the |, and the symbols ∈ and ℝ.
Finally, you don't need Cantor's diagonalization for a proof. It is easy(1) to show that, for any real x<1, x+(1-x)/2 is real, less than one, and strictly greater than x.
Alternatively, assuming 0<x<1, write x in decimal, and increase the first digit less than nine by one to get a larger real less than one. There is such a digit because 0.9999999… is not less than one.
(1) depending on how deep you want to descend into the foundations of mathematics.
Also, any notation will have to be learned. Concise ones a bit more, but there is nothing one can do about that.
And that perfectly clear notation requires you to learn quite a bit by memorization, such as the meaning of those {} brackets, the |, and the symbols ∈ and ℝ.
Finally, you don't need Cantor's diagonalization for a proof. It is easy(1) to show that, for any real x<1, x+(1-x)/2 is real, less than one, and strictly greater than x.
Alternatively, assuming 0<x<1, write x in decimal, and increase the first digit less than nine by one to get a larger real less than one. There is such a digit because 0.9999999… is not less than one.
(1) depending on how deep you want to descend into the foundations of mathematics.