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Tweets by jtbrazas-

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# Category Archives: Fundamental group

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

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 commutator, Frechet-Uryshon Fan, free group, infinite earring group, infinite earring space, locally free group, non-locally compact space, quasitopological group, quotient space, sequential space, topological group
1 Comment

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

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

## 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 compact-open topology, fundamental group, quasitopological fundamental group, quotient map, quotient topology, universal property of quotient maps
3 Comments