Epislon & Delta for Open / Not Open Sets

SteveL27

If you can describe S in words, it will be very helpful.

S consists of all the points in the plane except for ___________.

Now if P is a point of S, can we find a little circle around P that does not contain any points of ____________?

thrill3rnit3

Hm, I think it would be nice to have an systematic approach to this proof.

What do we want to show? We want to show that for any point p (x,y) there exists a radius δ > 0 such that, for any point q, if the distance from p to q is less than δ,

ie. if l p - q l < δ where l l represents the [ Euclidean? ] metric,

then q belongs to S.

The "there exists a radius δ > 0" is a very important part. It means that you have to come up with a delta, which depends on the chosen point p, so that the above holds.

Don't try to write down the "proof" yet. Make sure you understand what is going on. Can you come up with a delta such that the conditions are satisfied?

Look at Stephen Tashi's post [ #7 ]. He basically outlines what you need to do. Let us know if there is a part that you are having trouble with.