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.

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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. 

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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.

image

 

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.

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