Hi I have started a new subject this week called Topology, and it a bit harder than the stuff that I am used too, but anyway
We have metric space called where two and . Then I need to show that there exists open sets
which statifies that 1) 2) and 3)
Firstly I know that the definition of open set or ball is as follows
In other words: A set U is considered open if there for every element in the set is the center of an open ball of the set.
I also know from the metric subject field that if x,y are point in the set T and if they a then d(x,y) > 0.
So the way to show 1) and possible 2) isn't that to claim that for set to be open I must show that every point m_j on that set will be the center of an open ball?
I can see all the definitions that I need in my head I just need some assistance to connect them
On page 49 there is not exactly a theorem but it states
Let a be a point in and let r be a possitive number then th set of all points x in such that
is called an open n-ball of radius and center a, and I et that epsilon and delta are some sort measures which are used to determen what happens at a.
There is a definition on the same page which states that every interior point a of a set S can b surrounded by an "n-ball" and any set containing a ball with cente a is refered to as a neighbourhood of a.
So basicly my task is to show there exist a neighbourhood around m1 which only holds m1 and not m2?
there is one which states d(x,y) > 0 -> and you choose a small delta and bit larger epsilon and the bigger delta gets the smaller epsilon until it reaches m1 and thusly m1 does not contain m2?
I am not trying to sound stupid or anything but its just then I get these generalized assignments I crash and burn, but then there is assignment with numbers then everything is fine.
I don't suppose you could explain to me in a example on how is much proceed? then everything works much better
Also "m1 does not contain m2" makes no sense. m1 and m2 are points, not sets.