Separable and normal topologies

[cothranaimeel]

Three topological spaces are given below. Determine which ones are separable and which ones are normal.(Hint on the separability part: For one of the spaces it is easy to construct a countably dense set, for another space you can prove every infinitelycountable set is dense, and in the other space you can prove that every countabe set can not be dense.

a) X=R with the cofinite topology t1 = {U proper subset of R: R~ is finite is finite}
b) X=R with e co-countable topology, t2= {U proper subset of R: R~U is countable
c) X=R^2 with the Euclidean topology.

help...I have been working on topology problems all day

 

[jvicens]

and I'm stuck!

First, let's define separability and normality for those who may not be familiar with these terms:

- Separability: A topological space X is separable if it contains a countable, dense subset. A subset A is dense in X if every point in X is either in A or a limit point of A.
- Normality: A topological space X is normal if for any two disjoint closed subsets A and B, there exist open sets U and V such that A is contained in U, B is contained in V, and U and V are also disjoint.

Now, let's analyze each of the given topological spaces:

a) X=R with the cofinite topology t1 = {U proper subset of R: R~ is finite is finite}
This space is separable because we can construct a countably dense set by taking all the singleton sets {x}, where x is a real number. Every point in X is either in this set or a limit point of it, since every open set in this topology contains infinitely many points. However, this space is not normal because there exist disjoint closed sets that cannot be separated by open sets. For example, consider the closed sets A={0} and B={1}. Any open set containing 0 must also contain infinitely many points, and thus cannot be disjoint from B.

b) X=R with the co-countable topology, t2= {U proper subset of R: R~U is countable}
This space is not separable because every countable subset is not dense. For any countable set A, there exists a point x in X that is not in A or a limit point of A. For example, if A={1,2,3,...}, then x=0 is not in A or a limit point of A. However, this space is normal because for any two disjoint closed sets A and B, we can take U=R\A and V=R\B, which are open and disjoint, and contain A and B respectively.

c) X=R^2 with the Euclidean topology
This space is separable because we can construct a countably dense set by taking all the rational points in X. Every point in X is either in this set or a limit point of it, since every open set in the Euclidean topology contains infinitely many points. This space is also normal because it is a metric space, and all metric spaces are