Topology: How to formally write proofs?

Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

A simple example: Prove that

Another simple example: Prove that

thanks for your time.

Re: Topology: How to formally write proofs?

You have to use english, or else you would have to resort to symbolic derivations in some logic. See the examples of proofs in a topology book and you'll get the style.

Re: Topology: How to formally write proofs?

Re: Topology: How to formally write proofs?

something to help you proving:

A ⊆ B implies AUB = B.

the way we show two sets are equal is to demonstrate they contain the same elements. so if we start with A ⊆ B, we need to show that everything in AUB is in B, and that everything in B is in AUB.

now everything in B is always in AUB (by the definition of union). so the "hard part" will be showing that everything in AUB is in B.

you'd start like so. suppose x is in AUB = {x in T : x is in A or x is in B} (here, T is "some set" that A and B both belong to, our "universe of discourse").

so either x is in A, or x is in B.

if x is in B, then certainly x is in B.

on the other hand, if x is in A, then.....(you should use something about the relationship between A and B here, now)