Wild Topology

Recently, I ran into a situation where I had an inverse sequence of *planar Peano continua.

In the multipart series on topologies on fundamental groups, we’ve discussed the fundamental group with the quotient topology: .

In Part I, we discussed the operation , which takes in a group equipped with a topology (that does not necessarily interact nicely with the operations of the group) and outputs a topological group.

The main complicating factor with the quotient topology on the fundamental group is that it often fails to be a topological group.

In Part 1, I defined local n-connectivity, which says that a space has a basis of n-connected open sets, and the porperty, which says that for all , “small” maps from the -sphere contract using “small” homotopies of “relatively the same size.”

I’m going to take a little break from the topologized fundamental group series for a bit.

This is the last in a sequence of posts about the quotient topology.

In Part 1, I mentioned that one of the surprising things about the natural quotient topology on the fundamental group is that the resulting group with topology often fails to be a topological group.

In Part 1, I described the construction of , the fundamental group equipped with the quotient topology and some of the drama around failing to always be a topological group.

Next up for topologies on the fundamental group is what I’d consider the most “natural” one.