Baire Category Theorem

[@StratosFair]

This is one of my favourite theorems ever !! It's really insane the amount of seemingly unrelated stuff you can prove by cleverly applying Baire's category theorem !

[@brightsideofmaths]

Indeed, always fun to see how one can define the suitable things to apply Baire's theorem.

[@toxicore1190]

do you have some examples (preferably of combinatorial nature?)

[@brightsideofmaths]

New upload because there was the mistake in the beginning, not mentioning the openness of the dense sets. Thanks for the viewers that pointed that out :)

[@Ghetto_Bird]

I hope you will continue this series at some point, I thoroughly enjoyed it!

[@brightsideofmaths]

Which series? :)

[@Ghetto_Bird]

@@brightsideofmaths Functional analysis 🤗 I thought you weren't quite finished yet with this topic.

[@brightsideofmaths]

@@Ghetto_Bird I have the "Unbounded Operators" as a series that continues a lot of topics. I will produce more videos there :)

[@Ghetto_Bird]

@@brightsideofmaths Aha, good to know! I'll see you there 😉

[@elsa1569]

Can Baire's Category theorem be seen as a different version of the statement that a countable union of sets with measure zero also has measure zero, provided we have a suitable measure and provided we restrict our view to measure spaces?

[@brightsideofmaths]

I wouldn't bring measures into the game. It brings in confusion because here only the topology tells you about meagre and "fat" sets.

[@elsa1569]

@@brightsideofmaths ok, I see. Hmmm.. isn't it weird that having a dense subset of a space feels (at least for me) like having "almost" all elements of the space meaning that there is not "much" to add by closure...yet, this is not necessarily true, take for example rational numbers in the space of real numbers.. for every real number we can find a rational number no matter how closely around the real number we look (doesn't that sound like there should be a on-to map?), yet, there are "more" reals that are not rational than there are rational numbers.... 🤷🏼‍♀️

[@LillianRyanUhl]

Generally speaking, "negligible" sets from topology and measure theory don't play nice with one another. It's a classic theorem that there is a disjoint partition of ℝ into a set of Lebesgue measure zero and a meager set

[@StratosFair]

@@LillianRyanUhl any reference for that ?

[@spogel9981]

Many thanks for this video. One question remains for me: Can you say what the Aj in the application are?

[@brightsideofmaths]

Thank you very much! The wikipedia page also covers this example. However, maybe I make another video about this.

[@Hold_it]

Will there also be a dark side version for this video?

[@brightsideofmaths]

Not yet but it will come :)

[@Hold_it]

Ok, can't wait :)

[@sussybawka9999]

Let's go!!!

[@toxicore1190]

the naming in topology is just bad

[@brightsideofmaths]

And this is not even the craziest naming ;)

[@toxicore1190]

@@brightsideofmaths yes, I really dislike the countability and separation axioms

[@angelmendez-rivera351]

@@toxicore1190 Poor naming of concepts in mathematics, always due to outdated traditions, is easily one of the greatest obstacles to learning mathematics.

[@jeandetti6250]

First

[@brightsideofmaths]

Second!

[@brightsideofmaths]

We are doing ordinals here, aren't we?