The Sierpinski Space.
March 22, 2011
What’s the weakest separation we can have in a topological space?
Well, “no separation” is pretty weak. But this creates the trivial topology and that’s a bit boring. So let’s say this:
Definition: A topological space is or Kolmogorov if for every two points we have that there exists a neighborhood such that either and or and .
In other words, a space is if for every pair of points there is at least one open set which contains one and doesn’t contain the other. This is a pretty weak separation condition. Certainly, every Hausdorff space is , but there are ones which are even weaker which satisfy this condition. Let’s try to construct a really easy one.
Since a 1-element set is somewhat trivial, let’s begin with a 2-element set. Let’s say our set is . Now, by the definition of a topology, we need to be open. Since we want to have this space be , we should probably make an open set that contains but does not contain . So our open sets are:
which gives us a topology on this set (check this!). Moreover, this was, by construction, the smallest set such that this topology was neither discrete (as is not open) nor trivial. In addition this space is non-Hausdorff, so this gives us a nice example of a space which has a strictly weaker separation property than any Hausdorff space. In fact, this set has a name.
Definition: Let be given the topology above. Then we call this space the Sierpinski Space and it is denoted .
Let’s just state a few nice properties of this.
- This space fails to be Hausdorff and Regular, but it is Normal since there are no pairs of disjoint closed sets.
- This space is connected (check this!) and, in fact, it is even path connected. How do we define a path on this? (Hint: check to see if sending the first point of our path to 0 and the rest to 1 is continuous.)
- Trivially, this space is compact.
An interesting point arises when we consider convergence, though. We’ll finish the post with this consideration.
Take a sequence in . Then this sequence converges to 0. The proof here is trivial: the only open set containing 0 is the entire space, and so each of the points is in this open set for every . The question is then, when does this sequence converge to 1? A little bit of thinking will show you that if we have only finitely many 0’s in the sequence then the sequence will converge to 1. For example, does not converge to 1 (why not?) but it still converges to 0.
What does the sequence converge to? Are limits unique in this case? What is the weakest separation we can have to force the limits of a sequence to be unique?
I’ll end with a cute exercise: try to iterate our process of making the Sierpinski Space but use three points; say, , then let be the topology. What can we say about this space? What kind of limits do our sequences have?
Repeat this with points. Repeat this with countably infinite points. Now you have a neat topology on . What kind of things do sequences in this topology converge to?