[Topology] Accumulation points

[GatorPower]

Homework Statement

Given N = {1, 2, 3, ...} and En = {n, n+1, ...} (n in N) define a topology on N with T = {empty set, En}
Determine the accumulation points of {4, 13, 28, 37}, find the closure of {7, 24, 47, 85} and {3, 6, 9, 12, ...}. Also determine the dense subsets of N.

Homework Equations

x is an accumulation point of a set A if it is in the closure of A - {x}, or equivalently if every neighbourhood (nbd) of x intersect A in some point other than x.

Closure of a set = intersection of all closed sets containing the set, or simply cl(A) = A in union the set of accumulation points of A.

A subset A is dense if cl(A) = N

The Attempt at a Solution

Let A = {4, 13, 28, 37}. Closed sets in N is: N - En = {1, ..., n-1} for a given n (empty set for n approx infinity) . The closure of A will then be N - E38 since we then get cl(A) = {1,2, 3, 4, ..., 13, .., 28, .., 37} which complement is open. Then each of the points 1, 2, ..., 36 are accumulation points (not 37 since then the closure would be N - E37). This seems a bit.. strange, so please help me on this if this is totally wrong.

Determining the closure of the other sets is easy if the previous paragraph is correct.

Dense subsets is only {empty set} since cl(empty set) = N

Please look over this, looking forward to hearing from some more experienced topologists :)

 

[Dick]

I think you've got the right ideas. The closure of A = {4, 13, 28, 37} is N-E38 (or the interval [1,37]). But I don't think the closure of A-{37} is N-E37. What is it really? And I certainly don't think the closure of the empty set cl({}) is N.

 

[GatorPower]

I think you've got the right ideas. The closure of A = {4, 13, 28, 37} is N-E38 (or the interval [1,37]). But I don't think the closure of A-{37} is N-E37. What is it really? And I certainly don't think the closure of the empty set cl({}) is N.

My bad, A-{37} would be closed with N - E29.

Closure of empty set, hmm.. N - empty set = N which is open and hence the empty set is clopen and not dense in N.

So, I'm looking for a set which closure is E1 = N. But the closure of a set is the intersection of all closed sets containing the set, and the only set containing N is N itself!

So, the accumulation points are 1,..,36 and the dense subset is only N. Correct?

 

[Dick]

My bad, A-{37} would be closed with N - E29.

Closure of empty set, hmm.. N - empty set = N which is open and hence the empty set is clopen and not dense in N.

So, I'm looking for a set which closure is E1 = N. But the closure of a set is the intersection of all closed sets containing the set, and the only set containing N is N itself!

So, the accumulation points are 1,..,36 and the dense subset is only N. Correct?

Do one of the other exercises they gave you before you jump to conclusions about dense sets. What's the closure of {3, 6, 9, 12, ...}. Notice the '...' indicates it has a infinite number of elements.

 

[GatorPower]

The closure of {7, 24, 47, 85} would be {1,...,84} since N - cl({7, 24, 47, 85}) then equals E85 which is open.

Now on to the closure of {3, 6, 9, 12, ...}. Then the closure would be {1, ...} which would be equal to N.

If this is correct I would think that any subset of infinite cardinality is dense?

 

[Dick]

The closure of {7, 24, 47, 85} would be {1,...,84} since N - cl({7, 24, 47, 85}) then equals E85 which is open.

Now on to the closure of {3, 6, 9, 12, ...}. Then the closure would be {1, ...} which would be equal to N.

If this is correct I would think that any subset of infinite cardinality is dense?

Sure. Any infinite subset will intersect ALL open sets. That's an alternative definition of dense.

 

[GatorPower]

Thanks a lot for the help!