Author Archives: Jeremy Brazas

Topologized Fundamental Groups: The Quotient Topology Part 4 (Subgroup Classifications)

Posted on October 7, 2023 by Jeremy Brazas

This is the last in a sequence of posts about the quotient topology. This one is about how the topological structure of can be used to classify certain generalizations of covering maps for locally complicated spaces. Sometimes it still amazes … Continue reading →

Posted in Algebraic Topology, core of a subgroup, Fundamental group, Generalized covering space theory, Griffiths twin cone, harmonic archipelago, Quasitopological groups, quotient topology, semicovering, topological fundamental group, Uncategorized | 7 Comments

Topologized Fundamental Groups: The Quotient Topology Part 3 (Why isn’t it always a topological group?)

Posted on April 18, 2023 by Jeremy Brazas

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 fact, I’d say it’s usually … Continue reading →

Posted in compact-open topology, earring group, Free groups, Fundamental group, quotient topology, topological fundamental group, Topological groups, Uncategorized | Tagged , , , , , , , , , , | 5 Comments

Topologized Fundamental Groups: The Quotient Topology Part 2 (Discreteness)

Posted on February 13, 2023 by Jeremy Brazas

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. In this second post, I plan to connect back to … Continue reading →

Posted in Fundamental group, Uncategorized | Tagged , , , | 4 Comments