Recent content by blahblah8724

For \alpha = (1+ \sqrt{-3})/2 \in ℂ and R = \{ x +y\alpha \, | \, x,y \in Z \}. How would you verify that R is a subring of ℂ? Everytime I multiply two 'elements' of R to check closure I get the negative complex conjugate of \alpha, I think I'm doing something wrong... Thanks!

Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?

Surely the union of those two halves don't make the entire space as they miss out the point 1?

How about the subset (0,2) where the two halves are (0,1) and (1,2)? So the closure would be [0,1] and [1,2] which intersect at 1?

But how could you possibly go about proving that there are NO nonempty subsets

Do you mean larger than T?

But one of the questions in my example sheet said think of an example of a disconnected subspace T of a topological space S for which there are no nonempty subsets A,B of T such that A\cup B = T but \bar{A} \cap \bar{B} = ∅ Surely if \bar{A} = A = A^o then \bar{A} \cap \bar{B} = A \cap B =...

For a subset which is both closed and open (clopen) does its closure equal its interior?

If the closure of a space C is connected, is C connected?

I am told that the interval (a, ∞) where a \in (0, ∞) together the empty set and [0, ∞) form a topology on [0, ∞). But I thought in a topology that the intersection if any two sets had to also be in the topology, but the intersection of say (a, ∞) with (b, ∞) where a<b is surely (a,b) which...

Just to help my understanding... 1) Can you have an abelian subgroup of a group which isn't normal? 2) Can you have a normal subgroup which isn't abelian? Cheers!

I have a question where it says prove that G \cong C_3 \times C_5 when G has order 15. And I assumed that as 3 and 5 are co-prime then C_{15} \cong C_3 \times C_5 , which would mean that G \cong C_{15} ? So every group of order 15 is isomorohic to a cyclic group of order 15...

Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties? So for example for a group G with order 15 to show that G \cong C_3 \times C_5 would I just have to define all the possible transformations to define the isomorphism...

But surely if again we use additive notatoin with C_9=\{0,1,2,3,4,5,6,7,8\} then f:C_9 \to C_3 \times C_3 with f(0) = (0,0), f(1) = (0,1), f(2) = (0,2), f(3) = (1,0), f(4) = (1,1), f(5) = (1,2), f(6) = (2,0), f(7) = (2,1), f(8) = (2,2) is as isomorphism?

Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?