The Sierpinski Space.

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.

Counter-Examples: The Particular Point Topology.

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 , we let 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 .  Convince yourself that this is, in fact, a topology by going over the definition of a topology. 

Counter-examples: The Line with Two Origins.

This counter example is pretty cute, I’m not gonna lie.  This is a great example to use if your class doesn’t believe that a sequence could converge to two different values if the space is non-hausdorff.  The drawing below should be enough to give you some intuition as to what this example is.

Essentially, we’re going to make a real line that has "two origins."  There’s a number of good constructions for this, but this is the one that I like the best:

Take the set union a single point which we’ll call .  Now, let’s specify a basis for this topology.  For every with :

• If does not contain the origin, then we include it into the basis.
• If contains the origin, then we put into the basis.

In other words, every open set that contains the origin also contains . 

 

This gives us something that kind of looks like the picture above, but we aren’t able to topologically distinguish the point and the origin.

First, this has many of the same properties of the real line.  It is second countable (it has a countable basis), it is non-compact, and every point is homeomorphic to a one-manifold.

Unlike the real line, it is non-hausdorff.  Specifically, we cannot separate the two origins.  Another really sweet property is that there is a sequence which converges, but not to a unique limit.  For example, take the sequence

This will converge to , of course, but it also converges to .  Thus this sequence doesn’t have a unique limit. 

Counter-examples: The Topologist’s Sine Curve.

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

for every .  We also include the vertical line at from .  The reason for this is that the closure of the image of 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.