Does Cocountable Topology Affect Countable Local Bases?

[zpconn]

Homework Statement

Let X be a set equipped with a topology tau1, and let tau2 be the cocountable topology in which a set V in X is an open set if V is empty or X - V has only finitely or countably many elements. Consider the topology tau consisting of all sets W in X such that for each point p in W there exist subsets A and B of X containing p such that A is open in tau1, B is open in tau2, and the intersection of A and B is a subset of W.

If (X, tau1) has a countable local base at a point p in X, then under what conditions does (X, tau) have a countable local base at p?

Homework Equations

The Attempt at a Solution

I am truly at a total loss on how to do this. I've been messing around with this problem for a long time and am just not seeing the solutions. At this point, I honestly don't have any work worth showing. I don't know if I'm just having a brain meltdown right now or if this is an unexpectedly hard problem.

 

[mighty2000]

Thank you for your post. It appears that you are struggling with this problem and are in need of some assistance. I am happy to help you work through this problem. Let's break it down step by step.

First, let's clarify the definitions of the topologies involved. Tau1 is the given topology on X, and tau2 is the cocountable topology. The topology tau is defined as the collection of all sets W in X such that for each point p in W, there exist subsets A and B of X containing p such that A is open in tau1, B is open in tau2, and the intersection of A and B is a subset of W.

Now, let's consider the statement that (X, tau1) has a countable local base at a point p in X. This means that there exists a countable collection of open sets in tau1, denoted by B, such that for any open set U containing p, there exists a set V in B such that p is contained in V and V is a subset of U. In other words, B is a countable collection of open sets that "cover" p.

Next, we need to determine under what conditions (X, tau) has a countable local base at p. This means that there exists a countable collection of open sets in tau, denoted by C, such that for any open set U containing p, there exists a set W in C such that p is contained in W and W is a subset of U. In other words, C is a countable collection of open sets that "cover" p.

To find these conditions, we need to consider the definition of tau and how it is related to tau1 and tau2. Notice that tau contains sets that are open in tau1 and tau2, and the intersection of these sets is a subset of W. This means that for any set W in tau, there exists a set A in tau1 and a set B in tau2 such that W is a subset of A intersect B.

Now, let's return to the statement that (X, tau1) has a countable local base at p. This means that there exists a countable collection of open sets in tau1, denoted by B, such that for any open set U containing p, there exists a set V in B such that p is contained in V and V is a