Category Archives: Group theory

Shape injectivity of the earring space (Part I)

Posted on July 31, 2019 by Jeremy Brazas

One of my posts where I did some substantial hand-waving is my original post on the fundamental group of the earring space. I wrote about how to understand and work with this group, but I never gave a proof of … Continue reading →

Posted in earring space, Free groups, Fundamental group, Inverse Limit, Shape homotopy group, Shape theory, Tree | Tagged , , , , , | 9 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

Homotopically Hausdorff Spaces (Part II)

Posted on March 18, 2017 by Jeremy Brazas

In my post homotopically Hausdorff spaces (Part I), I wrote about the property which describes the existence of loops that can be deformed into arbitrarily small neighborhoods but which are not actually null-homotopic, i.e. can’t be deformed all the way back … Continue reading →

Posted in Algebraic Topology, earring space, Fundamental group, Group homomorphisms, harmonic archipelago, homotopically Hausdorff, Homotopy theory | Tagged , , , | 2 Comments