I am working on Topological Complexity of robot motion planning. I am looking for a connected CW-complex which is not metrizable. I have found that:
Proposition 3.8. A connected CW complex $X$ is metrizable if and only if it is locally finite.
A CW complex is locally finite if each cell is disjoint from all but finitely many cells of $X$.
Proposition 10.1.8. A CW complex is locally compact iff it is locally finite.
Proposition 1.5.12 A locally finite and connected CW-complex $X$ is countable.
So I have came up with the following example:
A graph with vertices the irrational numbers and edges the intervals between two consecutive ones.
According to the last proposition it shouldn't be locally finite (and intuition also says the same here).
What do you think about the example? Is there an easier one out there that I am missing?
Best Answer
There is no such thing as "consecutive irrational numbers". What would the number that comes after $\sqrt{2}$ be? Between two irrational numbers there is always another one... Your CW complex doesn't exist.
For an example of a connected, uncountable CW complex, you can for example take this one: it has a single 0-cell, and an uncountable number of edges joining the 0-cell to itself. In other words, the wedge sum of an uncountable number of circles. It's clearly connected and uncountable, and as you can see it's not locally finite because the vertex meets an infinite number of cells.