Adventures in Point Set Topology

This semester I’ve taken a course covering, again, introductory point set topology. This is a fairly standard and basic subject to cover in an undergraduate mathematics course. There is something fascinating about point set topology; it invites conjectures and then dashes those conjectures in satisfying and clever ways.

I have, for today only, been wondering about how to develop intuition regarding regular and normal Hausdorff spaces. Ignoring the more complex separation axioms here, let’s just say that a regular space has each disjoint pair of a point and a closed set are separable by open sets. A normal space is similar but for disjoint pairs of closed sets.

It was initially unclear to me why it would be the case that regular and normal spaces are in fact separate things. It seems that it would be difficult to construct a topology such that one could separate points and closed sets, but not closed sets and closed sets.

It occurred to me that in some sense, a normal space is more infinitely divisible than a hausdorff or regular space. In that I can fit the boundary of an open set into tighter spaces.

In the end I couldn’t construct a regular space that was not normal in a short enough time to justify thinking about a non-assessable problem, so I looked it up. I’ll provide some examples at the end of this rant.

An interesting fact came up while discussing the problem with some classmates. They could not come up with examples either, however one suggested taking a set and declaring each closed set to be an open set.

This is not a solution, but it has the interesting property that since a (non-finite) regular space must be $latex T^1$, every T^1 space has every singleton point as a closed set. Therefore such a condition on a T^1 space immediately yields the discrete topology. As this is the topology whose base is each singleton point.

This is odd, because to go from a relatively unstructured topology that is merely T^1 to the finest topology with such an innocuous condition is a bit counter-intuitive.

To segue into examples, my favourite example of a T^0 space is the topology given by all sets that contain a specific distinguished point x \in X, with X the space.. This is not T^1 because \{x \} is not closed, as it’s complement does not contain \{ x\}.

It is however T^0 as each pair of points admits an open set about one of them not containing the other.

Before giving some examples of regular and not normal spaces. The aim of giving these examples is to share the intuition I’ve been attempting to gain on these definitions. First here are some theorems, presented without proof

Munkres’s Topology book, p 200:

Theorem 32.1 Every regular space with a countable basis is normal.

An immediate corollary is that for finite sets, every regular topology is also a normal topology, and so the two are equivalent because, in general, a normal topology is also a regular topology.

So any desired example must be non finite, and must also not admit a countable base.

Details of specific examples are non-trivial, in this sense such topologies are somewhat artificial.

Due to time constraints, these will be added a later date.


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