## 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.

## Counter-Examples: The Particular Point Topology.

### November 3, 2010

I just used this counter-example, so I felt like I should share it with all of you guys.

The particular point topology is defined in the following way: given some space $X$, we let $p\in X$ be a distinguished (or particular) point.  It can be any point, really.  Then we let a set be open if it is the empty set, or if it contains $p$.  Convince yourself that this is, in fact, a topology by going over the definition of a topology.

## Counter-examples: The Topologist’s Sine Curve.

### October 8, 2010

After I learned about the topologist’s sine curve, I started using it almost immediately; it’s a really sweet example of a graph that is connected, but is not path connected or even locally path connected!  Let’s just jump right in and define it.

The equation for the topologist’s sine curve is

$f(x) = \sin(\frac{1}{x})$

for every $x\in (0,1]$.  We also include the vertical line at $x = 0$ from $-1\leq y \leq 1$.  The reason for this is that the closure of the image of $f(x)$ includes it.  This is easy to see if we notice that the curve goes up-and-down very quickly near 0.  It does take a bit of proving, but not much (it suffices to show every point on the vertical line we added is a limit point).

Now, let’s show a few properties that the topologist’s sine curve has.