Author Archives: Jeremy Brazas

Topological Group Reflections: turning a group with topology into a topological group, Part I

Posted on September 19, 2024 by Jeremy Brazas

The main complicating factor with the quotient topology on the fundamental group is that it often fails to be a topological group. More generally, if is a topological monoid and the quotient space is group by a relation that preserves … Continue reading →

Posted in Categorical Topology, quasitopological groups, quotient topology, reflection functor, Topological groups, Uncategorized | 4 Comments

Local n-connectivity vs. the LC^n Property, Part 2

Posted on October 17, 2023 by Jeremy Brazas

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 … Continue reading →

Posted in Algebraic Topology, Examples, General topology, Higher Homotopy groups, Homotopy theory, locally n-connected spaces, locally path connected, Peano continuum, Uncategorized | Tagged , , , , , , | Leave a comment

Local n-connectivity vs. the LC^n Property, Part I

Posted on October 8, 2023 by Jeremy Brazas

I’m going to take a little break from the topologized fundamental group series for a bit. That’s a long series and a distraction might be nice. Right now, I’m going to share a little about a property I’ve been running … Continue reading →

Posted in Dendrite, first uncountable ordinal, Higher Homotopy groups, locally path connected, one-dimensional spaces, path components | Tagged , , , , , | 1 Comment