## 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 $X$ is $T_{0}$ or Kolmogorov if for every two points $x,y\in X$ we have that there exists a neighborhood $U$ such that either $x\in U$ and $y \notin U$ or $y\in U$ and $x\notin U$.

In other words, a space is $T_{0}$ 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 $T_{0}$, 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 $\{0,1\}$.  Now, by the definition of a topology, we need $\emptyset, \{0,1\}$ to be open.  Since we want to have this space be $T_{0}$, we should probably make an open set that contains $1$ but does not contain $0$.  So our open sets are:

$\tau = \{\emptyset, \{1\}, \{0,1\}\}$

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 $\{0\}$ 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 $\{0,1\}$ be given the topology $\tau$ above.  Then we call this space the Sierpinski Space and it is denoted ${\mathbb S}$

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 $\{a_{n}\}$ in ${\mathcal S}$.  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 $n\in {\mathbb N}$.  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, $0,1,0,1,0,1,0,\dots$ does not converge to 1 (why not?) but it still converges to 0.

What does the sequence $1,1,1,1,1,\dots$ 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, $\{0,1,2\}$, then let $\tau = \{\emptyset, \{2\}, \{1,2\}, \{0,1,2\}\}$ be the topology.  What can we say about this space?  What kind of limits do our sequences have?

Repeat this with $n$ points.  Repeat this with countably infinite points.  Now you have a neat topology on ${\mathbb N}$.  What kind of things do sequences in this topology converge to?