1. ## Interior points proof

Any hints or clues appreciated.

Here are my answers so far:

a) if A ∩ S is emtpy then ∅ ∈ ΩS

b) Since A ∈ ΩX(which means A is open), there exists r>0 s.t B(x,r) ⊂ U therfore x ∈ intxA.

c) Don't know

Suppose x ∈ (intxA) ∩ S....

d)?
e)?
f) U is an open set
g) All open sets can be expressed as a union of balls
h) ?

2. Frankly I am tired of dealing with the way you post questions.
It is a waste of time on your part as well as our part.
This current question is unreadable at best.
Why do you think that any of us ought to know what $\Omega S$ means? What is $A~?$
Moreover, you have posted an image that actually cuts-off the given in the question. Do you fear being caught cheating by one of the net-search programs that check for copying?

http://www.mathhelpforum.com/math-he...ial-19060.html
Why not learn to post in symbols? You can use LaTeX tags
Here is an example.
[tex]\Omega S[tex] gives $\Omega S$

3. Sorry thought that this notation was standard...
ΩX = The set af all open sets in X
ΩS = The set af all open sets in S

These are tutorial questions, not any assignment questions(we only write tests), so cheating not an issue, there are no marks invovled... The top part which is cut-off.. has nothing to do with the question.