Two simple questions in topology.

1. Dec 2, 2007

MathematicalPhysicist

just want to see if i got these:
1.let U be open in X and A closed in X then U-A is open in X and A-U is closed in X.
2. if A is closed in X and B is closed in Y then AxB is closed in XxY.

my proof:
1.
A'=X-A which is open in X
X-(A-U)=Xn(A'U U)=A'U(U) but this is a union of open sets and thus it's open thus:
A-U is closed.
U'=X-U which is closed in X
X-(U-A)=Xn(U'UA)
but this is an intersection of closed sets and thus it's also closed.
which means that U-A is open.

2.
let's look at:
XxY-AxB=((X-A)xY)U(Xx(Y-B))
now X-A is open in X and Y is open in Y so (X-A)xY is open in XxY the same for Xx(Y-B) so we have here that it equals an arbitrary union of open sets so it's also an open set, thus AxB.

2. Dec 2, 2007

JasonRox

The only problem I see is that for question 2, you're leaving the reader to fill in some blanks like how you're going to solve the problem and the conclusion.

Start like this...

I will show that AxB is closed in XxY by showing XxY-AxB is open in XxY.

See what I'm saying?

I hate writing proofs and I suck at writing them, but I do know how to write them properly if I really really had to.

3. Dec 3, 2007

MathematicalPhysicist

those are minute details, and usually in exams i do write the proofs properly, this is only a few questions from munkres (the easy ones) that im solving for myself.