Recent content by emptyboat

Yes. I understand countable meaning and subtle difference between rational numbers and real numbers. I pick a random rational number as 12/99 and if order can be given it's countable. But if I pick π, I cannot give order to that number. So it's uncountable. Thanks!

12/99 is correspond to ...1212121212, but there is not such a natural number. But I think if your logic is true, rational numbers are also uncountable. In the case of rational numbers, countable means like that, it shows a possibility of one-to-one correspondence. But in the case of real...

If natural numbers have countable digits, and every dicimal also has countable digits. (Because it's mirror refection with zero in center). Natural numbers are countable and natural numbers are greater than digits of natural numbers, so digits of natural numbers are countable, and also every...

It's very simple. (mirror refection with zero in center) Every sigle-digit decimals correspond to every single-digit natural numbers(9 pieces). (1-0.1, 2-0.2, 3-0.3, ..., 9-0.9) Every two-digits decimals correspond to every two-digit natural numbers(90 pieces). (10 - 0.01, 11 - 0.11, 12-0.21...

If I 1 2 3 4 ... turns into 2 12 103 1004 I can go on this process. Natural numbers have the potential to be infinitely long.

Your logic is like this. 1 - 1 2 - 2 3 - 3 ... 10 - 10 11 - 11 ... 123-123 ... Assume that this list contains all natural numbers. Now add 1 to the ones place of the first number.(1+1=2), add 1 to the tens place of the second number(0+1=1), add 1 to the hundreds place of the third...

I think that real number is countable. Because there is one to one correspondence from natural numbers to (0,1) real numbers. 0.1 - 1 0.2 - 2 0.3 - 3 ... 0.21 - 12 ... 0.123 - 321 ... 0.1245 - 5421 ... I think that is a one-to-one corresepondence. Any errors here?

Thanks a lot, arkajad. I understand it. if only one coordinate is different, they are disjoint.

Hello, everyone. Theorem) If each space Xa(a∈A) is a Hausdorff space, then X=∏Xa is a Hausdorff space in both the box and product topologies. I understand if a box topology, the theorem holds. but if a product toplogy, I do not understand clearly. I think if there are distinct points c,d in...

Yes, I got it. You mean arbitrary large family is admittable. Thanks a lot !

normal extension - an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. (wikipedia) I have a question about the 'family of polynomials' it says the family should be arbitarary large? if E be an algebraic closure of Q(rational...

Thanks a lot, mrbohn1. I understand it. I think 'Cyclic' is deduced by 'primitive element Theorem'.

Hello... I have a question about the solvable group. I read a Fraleigh's 'A first course in abstract algebra', there is a question in sec56, exercise3. It says "The Galois group of a finite extension of a finite field is solvable." is true... I can't figure out why it's true. I think this...

Yes, I mean that. So, there was no clear answer about that?

Geometric Constructible Numbers... Hi, everyone. I have a question about geometric constructible numbers. I know that "if 'a' is constructible then [Q(a):Q]=2^n." But I heard that its inverse is not true. I want some counter examples about the inverse statement. (I have checked by googling 'i'...