Tag Archives: free group

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

What is an infinite word?

Posted on August 13, 2018 by Jeremy Brazas

In this post, we’ll explore the idea of non-commutative infinitary operations on groups, that is, multiplying together infinitely many elements in a group. This idea arises very naturally in “wild” or “infinitary” algebraic topology. In fact, lately, this has been … Continue reading →

Posted in Group theory, Infinite Group Theory, Order Theory | Tagged , , , , , , , , | 1 Comment

The Harmonic Archipelago Group is not Free

Posted on May 18, 2014 by Jeremy Brazas

In recent posts, I’ve been writing about the behavior of fundamental groups of the most fundamental “wild” spaces. We did a decent amount of work in a two-part post to convince ourselves that the fundamental group of the earring space … Continue reading →

Posted in earring group, earring space, Free groups, Fundamental group, Group homomorphisms, harmonic archipelago | Tagged , , , , , , , , | 2 Comments