(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X = {1, 2, 3, 4, 5} and V = {{2, 3, 4}, {1, 4, 5}, {2, 4, 5}, {1, 3}} be a subbasis of a topology U on X.

a) find all dense subsets of the topological space (X, U)

b) let f : (X, U) --> (X, P(X)) be a mapping defined with f(x) = x (P(X)) is the partitive set of X). Is f continuous?

3. The attempt at a solution

a) Since V is a subbasis, then the family of all finite intersections of the elements of V form a basis B. So, B = {{2, 3, 4}, {1, 4, 5}, {2, 4, 5}, {1, 3}, {1}, {3}, {4}, {2, 4}, {4, 5}, {0}}. A subset D of X is dense in X iff every open set in X intersects D. So, D = {1, 3, 4}. To make sure this is right, we can see if Cl(D) = X. The closed sets in X are {{1, 2, 3, 5}, {1, 3, 5}, {1, 2, 3}, {2, 3, 4, 5}, {1, 5}, {2, 3}, {1, 3}, {2, 4, 5}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}}. The intersection of all closed sets containing D is {1, 2, 3, 4, 5} (it's the only set containing D, btw), so Cl(D) = X.

b) A mapping f between topological spaces is continuous iff the preimage of every open set in the codomain is open in the domain. For all sets in P(X) which are contained in U, this is obvious. If we take a set in P(X) (for example {2, 3}) which is not in U, then its preimage is the empty set, and hence is open in U. So, f is continuous.

Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Another topology on the naturals

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**