Hausdorff Subtopology: Is Second Countability Preserved?

Hey guys! Let's dive into a fascinating question in general topology that's been floating around: If we have a Hausdorff space that's also second countable, and we tweak its topology to make it coarser while still keeping it Hausdorff, does this new, coarser topology remain second countable? It's a bit of a brain-bender, so let's break it down.

Understanding the Key Concepts

Before we get into the nitty-gritty, let's quickly recap some key topological concepts to make sure we're all on the same page. This will help us better grasp the question and the potential answers.

• Hausdorff Space: A topological space is called Hausdorff (or T2) if for any two distinct points, we can find disjoint open neighborhoods for each. In simpler terms, you can always separate two different points with open sets that don't overlap. This property is crucial for many constructions and theorems in topology, as it ensures a certain level of “niceness” in the space.
• Second Countable Space: A space is second countable if its topology has a countable base. A base for a topology is a collection of open sets such that any open set in the topology can be written as a union of sets from the base. So, a second countable space has a “manageably small” collection of open sets that can generate the entire topology. This property is quite significant, as second countability implies several other important properties, such as separability and the Lindelöf property.
• Coarser Topology: Given a set $X$, a topology $\tau_2$ is coarser than another topology $\tau_1$ on $X$ if every open set in $\tau_2$ is also open in $\tau_1$. In other words, $\tau_2$ has “fewer” open sets than $\tau_1$. Think of it like this: $\tau_1$ is a more refined topology, capable of distinguishing more points and sets, while $\tau_2$ is a more “blurred” version.

Now, with these definitions fresh in our minds, let's rephrase our central question: If we start with a “nice” space (Hausdorff and second countable) and make its topology “less refined” while keeping it Hausdorff, does the property of having a countable base survive this change?

The Main Question: Second Countability and Coarser Topologies

Okay, so let’s really sink our teeth into the main question here: If we've got a topological space, let's call it $(X, \tau_1)$, and it's both Hausdorff and second countable, that’s a pretty sweet starting point. Now, imagine we mess with its topology a bit, making it coarser – meaning we're essentially removing some of the open sets. We end up with a new topology, $\tau_2$, on the same set $X$. But here’s the kicker: we're making sure this new space $(X, \tau_2)$ stays Hausdorff. The burning question is: does this new topology $\tau_2$ automatically inherit the second countability from the original $\tau_1$? Is $\tau_2$ guaranteed to have a countable base just because $\tau_1$ did?

This is where things get interesting. Intuitively, you might think, “Well, if we're just removing open sets, shouldn't it still have a countable base?” But topology, as we know, can be full of surprises. The key here is to really dig into what second countability means. It's not just about the number of open sets; it’s about whether you can find a countable collection of open sets that can generate all the other open sets through unions. And when you make a topology coarser, you're changing the relationships between sets in a way that might mess with this delicate balance.

So, let's try to think of some scenarios. What kind of changes could we make to a topology that would preserve the Hausdorff property but potentially lose second countability? Are there some