Category Archives: General topology

Topologized Fundamental Groups: The Quotient Topology Part 1

Next up for topologies on the fundamental group is what I’d consider the most “natural” one. It’s almost certainly the topology you’d most often get if you asked random topologists on the street to construct one for you. This is … Continue reading

Posted in Algebraic Topology, compact-open topology, Fundamental group, Quasitopological groups, quotient topology, Topological groups, Uncategorized | Tagged , , , , , | 1 Comment

Homotopically Reduced Paths (Part III)

In this last post about reduced paths, I’m going to work through the details of one of the most useful results in wild topology. Writing this post helped me work out my own way of proving this result and hopefully … Continue reading

Posted in Algebraic Topology, Dendrite, Fundamental group, Fundamental groupoid, Homotopy theory, Inverse Limit, one-dimensional spaces, reduced paths | Leave a comment

Shape injectivity of the earring space (Part II)

This is the sequel to Shape injectivity of the earring space (Part I) We’re on our way to proving the canonical homomorphism from the earring group to the inverse limit of free groups is injective. Part I was mostly dedicated … Continue reading

Posted in Dendrite, earring group, earring space, Free groups, Fundamental group, Inverse Limit, Peano continuum, Shape theory, Tree | 3 Comments