2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.
5. Explain how we can think of the unit sphere (a subspace of IR^3 with its usual topology) as a subset of the Hilbert cube.
We'll prove that no two open setsare disjoint, which implies the fact that X is not Hausdorff (see attached).
First assume X is uncountable.
Let there exist an open set A, in X. Since A is open, XA is countable and closed. Since X is uncountable, A must also be uncountable, as XA is countable. ...
In a co-finite topology is a subset Hausdorff? Are compact sets closed? Explain how could we think of the unit sphere as a subset of a Hilbert cube.