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.
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 , 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.
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
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.